Magnon Bose–Einstein Condensation and Superconductivity in A

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Y. by Edited and Israel; Israel 91904, 91904, Jerusalem University, Hebrew The Physics, a lattice Volkov A. 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PHYSICS P The conduction electron Hamiltonian reads Hc = A C k,p=± h † Ep (k)ψ ψ , 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 ⊥ √ J ,t g stant is set to unity, and ψk,± = (ψk,1 ± ψk,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 conduction 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 † Shaded ovals represent the localized spin singlets in the VBS phase. (B) The where sr,α = ψr,α(σ/2)ψr,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 † † † H ≈ (a a )(ψ σ ψ + ψ σ ψ ) 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 † + † + − √ [ar (ψ σ ψr,− + ψ σ ψr,+) + 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. theoretical explanation for the observed non-Fermi liquid scaling Numerical Methods in YbAl3C3. For the numerical solution of the model, we use the DMRG The Model algorithm as implemented in the ITensor package (28), targeting We consider the Kondo–Heisenberg model on the two-leg ladder the low-lying states and their physical properties. The presence (Fig. 1A), governed by the Hamiltonian H = Hc + Hf + HK . Hc of gapless and near-critical modes requires a careful numeri- contains the single-particle dispersion of the conduction band, cal analysis to avoid a bias toward low-entangled states. To that Hf describes the interacting spin-1/2 local moments, and HK cor- end, we have monitored the convergence of DMRG results as responds to the Kondo coupling between the two. For Hf we a function of bond dimension, keeping up to 9,830 states for consider the Heisenberg model on a ladder (23), system sizes up to L = 76 rungs (see SI Appendix, section 6, for details). X X X z Hf = J⊥ Sr,1 · Sr,2 + Jk Sr,α · Sr+1,α − hgf Sr,α, Magnon BEC Transition r r,α r,α [1] We first consider the one-band case for fields below the BEC transition h < hc . As the second band is gapped, one can inte- where Sr,α denotes a vector of spin-1/2 operators at site r, chain grate the fermions out (SI Appendix, section 2). Ignoring the α = 1, 2. For J⊥ 6= 0 (24, 25) the ground state at h = 0 is adia- hard-core constraint, we find the leading terms (in JK ) to renor- batically connected to a direct product of singlets on each rung malize the bosonic spectrum. In particular, the critical field is and thus serves as a minimal model of the VBS state. This state reduced, does not break any symmetries of the model in Eq. 1; however, 2 Z kF at h > hc a transition to a state with a quasi–long-range ordered JK dk 1 hc (JK ) ≈ hc (0) − AFM state occurs (23) (see also below). Thus, the magnetic tk 8t⊥ 0 π 1 + 2 cos(k) field h allows us to tune the ground state of the local moments t⊥ from a quantum disordered (VBS) state to a more conventional J 2 Z Z kF − K dkdk 0 [3] symmetry-breaking one (AFM), in full analogy to the frustration 64π2 parameter proposed in refs. 8 and 9. In what follows, we set the g |k|>kF −kF 1 factor for the local moments to gf = 1. For analytic calculations, × , 2 k−k0 0 we concentrate on the regime J⊥ Jk in a finite h ∼ J⊥. In this Jk sin 2 − tk[cos(k) − cos(k )] regime, the two low-energy states on each rung are the singlet and the lowest lying triplet, whereas the other two triplet states where the expression is for the case of the bottom band being 2 are separated by an energy gap of size ∼ J⊥. The low-energy sec- partially filled and JK /t⊥ Jk is assumed (see SI Appendix for tor can be exactly mapped onto either hard-core bosons (denoted details). The top line of Eq. 3 derives from the transverse part as ar for an annihilation operator) or spinless fermions (denoted of the Kondo coupling (Eq. 2, bottom line), whereas the bottom as fr and referred to as spinons) (23, 26). In either representation line arises from the longitudinal part (Eq. 2, top line). an occupied site is equivalent to a rung in the triplet state, while Up to second order in JK we find that the total magneti- z z z P z z an empty site is a singlet along the rung. zation M = Mf + Mc ≡ r,αhSr,αi + hsr,αi vanishes for fields

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(Fig. (x n opt the compute and o various for , M τ τ ), c z simaginary is IAppendix SI nthe On L. perin appear h χ(π J J ⊥ c k (J h ahdln eit h etraiersl fEq. of result perturbative the depicts line dashed The . = ) [4] K SI 1, ≡ ) J k = iesaigo h pngap spin finite- the the of study we scaling transition, size the with associated exponents ical parameter Luttinger a and scaling euto Eq. of result determine h accurately to tion method unbiased identified this point in crossing single clear ing a a observe field at we cross where critical precisely the sizes with system of set at function exponent dent scaling dynamical that arbitrary implies a an this assume criticality we Near BEC 2. magnon the with able 5.Ls,coet rtclt,tefe emo eutipisa implies result fermion free the criticality, parameter to Luttinger value, close class Last, universality (5). magnon-BEC find predicted We the details). for 6, section of scaling finite-size xetdt edsrbdb h aeter Eq. theory same the of by ity described be to expected vector this wave ordering in the self-energy to Assuming equal at boson not are the divergences momenta these While have case. could two-band regime the transition in BEC ble magnon the of metal. class a a in at universality calculation the DMRG firming the to with close agreement field excellent find we 0.3, c (J opeieylct h rtclfield critical the locate precisely To oetmt h orlto eghexponent length correlation the estimate To enwageta h E rniinas ean sta- remains also transition BEC the that argue now We t hspoiiga cuaeetmt of estimate accurate an providing thus L, K ⊥ J ∆ h k ) h = c s χ(π c (J ∼ L n n udai decrease quadratic a find and 2C) (Fig. 1.5, (J h K z J = J = K K ) 2 K t a aihn cln ieso,adconsistent and dimension, scaling vanishing a has ) o oprsn oi iedpcstesaigprediction scaling the depicts line solid comparison, For 0.4). ) k ∼ ngo ulttv gemn ihtefree-boson the with agreement qualitative good in , h 3. = c si h n-adcs.Hne h universality the Hence, case. one-band the in as h L i.e., , c 1.0, ∆ β J (J with , k s L K h oosi h iiiyof vicinity the in bosons the , 2 J K hsfrhrcon- further thus 2D, Fig. in shown as ), safnto of function a as ∆ R K = h s lig (A) filling. 1/8 and 0.4, β 1 = L c uvsa rtclt (see criticality at curves hsrlto stse nFg 2B, Fig. in tested is relation This . 1 = z K versus ∆ 2) hc gives which (23), nteAMphase. 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PHYSICS class of the transition is unchanged, and corrections to Eq. 3 are while the 1D Hubbard model does not possess the VBS sub- subleading at weak coupling. system. In the case of the t − J model, the binding of holes is achieved due to the energy cost J⊥ of breaking a singlet bond SC in the Two-Band Case (37, 38). The difference from the results for the t − J model We now consider the properties of conduction electrons for the is that the VBS fluctuations are not completely local, espe- two-band case (Fig. 1B). We determine the emergent phases cially for h close to hc and no requirements on the magnitude using bosonization (26), with the low-energy excitations of the of t⊥ results (37). ± two bands being described with the real bosonic fields ϕσ (x) and The expectations above are confirmed by the DMRG results ± ϕρ (x) for the spin and charge sectors, respectively [each field in Fig. 3. First, to estimate the SC gap, we obtain the spin ± also has a canonically conjugate one, θσ,ρ(x)]. In the absence of gap ∆s , in Fig. 3A with a finite size scaling analysis of the z z a Kondo coupling, the low-energy excitations of each of these energy difference E(M = 1, L) − E(M = 0, L), between the z z fields are described as a Luttinger liquid with K = 1, u = vF ≡ ground states in the M = 1 and M = 0 sectors (see SI 2tk sin kF . Appendix for details). Indeed, we find a nearly vanishing spin Ignoring for the moment the aforementioned Zeeman split- gap ∆s ≈ 0 for weak Kondo coupling, Jk = 0.2, and a finite ting, integrating out the gapped hard-core bosons leads to two gap, ∆s ≈ 0.17, at larger Kondo coupling, Jk = 2.4. To fur- interaction terms ther characterize the above phases, in Fig. 3B, we investigate the c electron intraband spin-density wave (SDW), O ± = 2 Z √ √ SDW JK + − + − † † ± (ψ ± ψ )(ψr,1,↑ ± ψr,2,↑), and SC (defined above), O , HI ,± = 2 2 dx cos 2(θρ − θρ ) cos 2(ϕσ ∓ ϕσ), r,1,↓ r,2,↓ SC 8ε + − π α˜ kF ±kF order parameters, through their respective correlation functions [5] + D + † + E χD/S(r) = (OSC/SDW) (L/2)OSC/SDW(r) . At Jk = 0.2, we where the index +(−) refers to individual bands (see Fig. 1B), find that both order parameters fall off like a power law, akin to 1/α˜ is a high-energy cutoff, and ε + − is the boson disper- kF ±kF the decoupled free-electron limit. By contrast, in the spin gapped + − phase, Jk = 2.4, the SDW correlation decays exponentially, while sion at q = kF ± kF . Semiclassically, the terms in Eq. 5 cre- ate a pinning potential for the fields making their excitations the SC correlation remains quasi-long range. gapped. To assess their possible impact in the quantum regime, Next, in Fig. 3C, we examine the spin resolved momentum + † we performed one-loop renormalization group (RG) analysis. distribution of the bonding band, nk,↑ = hψk,↑ψk,↑i. The dis- The corresponding equations have been derived in ref. 31; we tribution evolves from a sharp Fermi edge, at small Jk , to take the perturbatively generated interactions, including Eq. 5, an incoherent distribution, characteristic of a Luttinger liquid, as the initial conditions and solve the equations numerically upon approach to the spin gapped phase. Notably, the Fermi (see SI Appendix, section√ 3, for details). We find that terms wave vector is unchanged throughout this transition, unlike ± proportional to cos(2 2ϕσ ) are generated and flow to strong the usual Kondo lattice model (39, 40), due to the number coupling together with the ones in Eq. 5, allowing for a semiclas- of spins per unit cell being even (i.e., two) (9, 41). Finally, sical analysis. The remaining terms lead to a renormalization of using the scaling of the bipartite entanglement entropy SE ∼ the Luttinger parameters. Minimizing the action including the c/6 log L for a conformal field theory with central charge c cosine terms, we find that out of four gapless fermion modes, (42), we find four gapless channels (c = 4) in the VBS metal. − + only the excitations of the total charge mode ϕρ + ϕρ remain In the superconducting spin-gapped phase, there is only a sin- gapless, corresponding to a state with power law correlations gle gapless channel (c = 1) corresponding to the total charge ± of the superconducting order parameter OSC(x) = ψ↑,±ψ↓,±(x) mode. These results are consistent with the expectations from ± weak-coupling RG. and the conduction electron density at 2kF . Furthermore, the equilibrium values of the gapped fields are such that the superconducting correlations are sign-changing between the bands, Interactions in the AFM Phase i.e., there is a π phase shift between the SC order parameters For h > hc (JK ), there is a finite density of spinons in the of the + and − bands, which is an analogue of d-wave pairing on fermionic representation of the Hamiltonian Eq. 1. The low- f the ladder. Additionally, the dominant velocity renormalization energy excitations around the spinon Fermi points at kF are is such that the SC correlations are stronger, i.e., decay slower, of the Luttinger liquid type (23) that can be described using ± than the 2kF density ones. These results resemble the case of bosonization. The Luttinger parameters K and u are known two-leg Hubbard ladders (32) that have d-wave superconducting functions of h (23). One can now rewrite the low-energy part and charge density wave correlations. Similar results have also of the Kondo coupling in terms of the bosonic fields resulting in been obtained for the Kondo–Heisenberg model away from the three contributions HK =∼ HV + HZ + HI , where we use ϕ(x) for VBS limit (with J⊥ = Jk) in zero magnetic field (33). the spinon fields. First, a velocity renormalization appears due to J Let us now discuss the interplay of the above effects and the H = √K R dx∂ ϕ∂ ϕ H V 2 2π2 x x σ. I , on the other hand, describes the Zeeman splitting EZ due to a finite JK . The Zeeman splitting EZ interaction between fermionic spinons and the Fermi sea can thwart SC (34, 35) unless the SC gap ∆SC is sufficiently larger Z then EZ (for an alternative discussion, see SI Appendix). As the JK f interactions are marginal, the gap is expected to be exponen- HI = 2 dx cos(2kF x − 2ϕ) 2 4(πα˜) tially small ∆SC ∼ vF /α˜ exp[−1/g], where g ∼ vF JK /(˜αε ± ), √ √ kF 2 [cos(2kF x − 2(ϕρ + ϕσ)) − cos(2kF x − 2(ϕρ − ϕσ))], whereas EZ ∝ JK . It follows that for infinitesimal JK , EZ dom- [6] inates, while at larger JK , ∆SC takes over, which represents f a Doniach-like competition between Zeeman splitting and SC. where kF is the Fermi wave vector of the spinons. As ϕ, ϕρ,σ This is similar to the competition between the Kondo coupling are slowly varying functions of x the integral averages to zero f and the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. except for two special cases, kF = kF , π − kF . If that is so, how- We note that the superconducting pairing due to VBS fluctu- ever, this term is relevant throughout the AFM phase,∗ and it ations differs from the conventional scenario of a spin density wave QCP (3), as well as the one in the 1D Hubbard (32, 36) or t − J * model. Unlike the SDW QCP (1, 3), the magnon BEC crit- The scaling dimension of this term is 1 − K + O(JK /vF ) with K < 1 throughout the AFM ical mode has z = 2 even without an interaction with fermions, phase (23).

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.2000501117 Volkov et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 n ihteZea-pi em ons h odto o Eq. for condition Eq. The in points. interaction Fermi 6 the Zeeman-split of the effect with the ing reconsider to need ∼ hnbt h F a n h etraieyidcdZee- induced perturbatively the and gap splitting AFM man in the linear both is then term this thus, and k For electrons. conduction the for splitting h Zeeman a to tionally uto lcrn,i saraoal supinfrarneof range a for g assumption reasonable a fields is it electrons, the duction neglected have we throughout While Factor g Electron Conduction green shaded the by shown is 1C. as Fig. interval, in regions an into broaden to ues finite At out. gapped magnets. itinerant in gap 3. Fig. okve al. et Volkov stable also is transition BEC magnon the finite that at show now We 3D and 2D to Extensions suppressed. additionally be to correlations gap spectral a in resulting order fields the bosonic of two of values the pins log(L). size system (χ of SDW logarithm (B) the 1/L. versus in entropy spectroscopy, function entanglement level linear Neumann energy a von the to of fit number scaling a occupation size is momentum finite line spin-resolved a Dashed from average. extracted moving gap, a spin using The (A) phase. gapped spin ducting h oe osdrdhr.Te2 xeso ftemdlin model the of extension 2D 1A The Fig. here. considered model the F c > o ob vrgdt eoi then is zero to averaged be to not J h rsneof presence The 0 6= ± K h c J g , stedmnn feto h od opig n we and coupling, Kondo the of effect dominant the is K ae leti nt ed,w xettesuperconducting the expect we fields, finite in albeit case, MGrslsfrtetobn ae for case, two-band the for results DMRG c M h ˜ ( ossso h adr ragdi ounrpattern columnar a in arranged ladders the of consists J m 4v f K z +1/2) F snneoee nteasneo od coupling, Kondo of absence the in even nonzero is nteoebn aefr2 n Detnin of extensions 3D and 2D for case one-band the in J v α ˜ ∼ F K hs ecnie u eut ocryoe othe to over carry to results our consider we thus, ; nec ae n ftetregpesmdsis modes gapless three the of one case, each In . J K 2 v F J K icse for discussed / H α ˜ Z J K = CD AB 1−K 1 − n xet aho hs pca val- special these of each expects one 2 iia oteoeigo h SDW the of opening the to similar , √ J K 2π J h R K < dxM n sprmtial larger parametrically is and h k c F f n hs emnsplitting Zeeman Thus, . f z k + = ,↑ ∂ g J ⊥ x ftebnigbn,for band, bonding the of k factor, ϕ = F σ 1, ± emlasaddi- leads term J J k K = g ˜ ( c m 4v 0.3, ftecon- the of +1/2) F t ⊥ 6 = start- , 0.1, π − t k L = = with Appendix for where oe nFg 1A. Fig. in model 4. Fig. assumptions, keep these to Under 2 dimension to 1, scaling parallel define be have momentum momenta We to the not. velocities and Fermi or energy Fermi the antiparallel of the being dimension with spots scaling cases the hot two the we for at interactions, dimension velocities the of scaling influence their the calculate assess To mode. critical the by connected points surface Fermi lcrn omtobnsa eoe o h od interac- Kondo Eq. to the addition additional in an now possesses term conduction before, states intraband low-energy the as the While onto bands 4). projected tion (Fig. two another form one of result- electrons the top stack on we 3D, layers for ing and coupling, interladder weak a with h xrcino h eta hrefo h cln ftebipartite the of scaling the from charge central the of extraction The (D) 44. AB 1.0, S n C(χ SC and ) h H q = = K intra lig o various for filling, 1/4 and 0.5, Detnin fthe of extensions 3D (B) and 2D (A) the of depiction Schematic ,± Q o eal) suigta h E cusfrbosons for occurs BEC the that Assuming details). for 0 = [e.g., D q J orlto ucin nalglgsae for scale, log–log a on functions correlation ) y y 4 (t) J √ K ls to close y 2 Q X 0 k,q (π = f (k, , π π H q)a , , K E intra π (M [ = [ε] ,± f ) (q)ψ (k, n3] nyfrin rudthe around fermions only 3D], in = J (Eq. J 2 1) K q) z Q k (t) ntemtli B n supercon- and VBS metallic the in ±,k+q † − k 0 ∼ 1 = ] E htsos r ope to coupled are spots) (hot swl stefis term first the as well as 7) z (M f NSLts Articles Latest PNAS (k, σ = hl h remaining the while , + ) uvsaesmoothed are Curves 0). z π ψ )(q 2 = ±,k y + nat(9 43). (29, intact − h L π .c = ) ., 4 (C 44. (see | f6 of 5 The ) [7] SI

PHYSICS of HK (Eq. 2) are irrelevant in d > 1. For the case of antiparal- stable in the presence of a metallic conduction band and retains lel [noncollinear] Fermi velocities at the hot spots, we find that its universality class, with the critical value of the field being the former has a scaling dimension (1 − d)/2 [−d/2] and the lowered. We have demonstrated that VBS fluctuations lead to latter 1 − d [−d]. This provides a strong indication that the BEC unconventional SC in the case of two partially filled bands. transition should retain its universality class in the one-band case. Finally, we have shown that the stability of the magnon BEC The above result implies that the quantum critical behav- transition extends to higher-dimensional versions of the model. ior is governed by the same theory as in the undoped case, Our results allow us to draw conclusions regarding field-induced i.e., that of a dilute z = 2 Bose gas. Interestingly, this predic- transitions in heavy-fermion materials with spins residing on frus- tion may be verified in YbAl3C3, where a VBS ground state of trated lattices. In particular, our results have lead us to a clear the Yb moments (44) is formed on a deformed triangular lat- interpretation of the observed criticality in YbAl3C3 and anchor tice (45), while the conductivity suggests metallic behavior (44). future studies of the phase diagrams and quantum criticality of Application of a magnetic field results in a quantum phase tran- frustrated Kondo lattices. It will be also interesting to extend sition (46), with the specific heat having been found to exhibit our theory to VBS states that may break a crystalline symme- a CV /T ∼ log(1/T ) behavior close to the QCP. Initially, this try as well as field-tuned VBS to AFM transitions that occur at behavior has been attributed to a possible non-Fermi liquid state fractional magnetization plateaus (19). formed due to Kondo coupling (46). However, our results sug- gest that the field-induced transition should not be affected by Data Availability. All data needed to evaluate the conclusions in Kondo coupling; indeed, for a z = 2, d = 2 BEC a log(1/T ) the paper are available in the manuscript and SI Appendix. divergence is expected (47, 48). Thus, our results allow one to interpret the observed anomalies as a signature of the stability of ACKNOWLEDGMENTS. The authors acknowledge useful discussions with D. T. Adroja, M. Aronson, A. Auerbach, P. Coleman, P. Goswami, G. Kotliar, the magnon BEC transition in metallic systems. E. J. Konig,¨ S. Parameswaran, S. Shastry, and Q. Si. S.G. and J.H.P. were supported by Grant 2018058 from the US–Israel Binational Science Foundation, Discussion and Conclusions Jerusalem, Israel. S.G. and J.H.P. performed part of this work at the Aspen In this work we have studied quantum critical properties and Center for Physics, which is supported by NSF Grant PHY-1607611. P.A.V. acknowledges the support by the Rutgers University Center for Materials phases of a 1D frustrated Kondo lattice with a nonmagnetic VBS Theory Postdoctoral fellowship. DMRG calculations were performed using state in magnetic field. We have shown that the field-induced the ITensor package (28). Numerical computations were carried out at Intel magnon BEC transition that occurs in the insulating limit is Labs Academic Compute Environment.

1. H. Lohneysen,¨ A. Rosch, M. Vojta, P. Wolfle,¨ Fermi-liquid instabilities at magnetic 24. S. R. White, R. M. Noack, D. J. Scalapino, Resonating valence bond theory of coupled quantum phase transitions. Rev. Mod. Phys. 79, 1015–1075 (2007). Heisenberg chains. Phys. Rev. Lett. 73, 886–889 (1994). 2. S. Sachdev, B. Keimer, Quantum criticality. Phys. Today 64, 29–35 (2011). 25. D. G. Shelton, A. A. Nersesyan, A. M. Tsvelik, Antiferromagnetic spin ladders: 3. D. J. Scalapino, A common thread: The pairing interaction for unconventional Crossover between spin s=1/2 and s=1 chains. Phys. Rev. B 53, 8521–8532 (1996). superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012). 26. T. Giamarchi, Quantum Physics in One Dimension (Clarendon Press, 2003). 4. P. Coleman, C. Pepin,´ Q. Si, R. Ramazashvili, How do fermi liquids get heavy and die? 27. R. Carlin, Magnetochemistry (Springer, 1986). J. Phys. Condens. Matter 13, R723–R738 (2001). 28. ITensor Library, Version 3.0. http://itensor.org (2019). Accessed 1 June 2019. 5. S. Sachdev, Quantum magnetism and criticality. Nat. Phys. 4, 173–185 (2008). 29. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, 1999). 6. S. Wirth, F. Steglich, Exploring heavy fermions from macroscopic to microscopic length 30. R. Chitra, T. Giamarchi, Critical properties of gapped spin-chains and ladders in a scales. Nat. Rev. Mater. 1, 16051 (2016). magnetic field. Phys. Rev. B 55, 5816–5826 (1997). 7. Q. Si, F. Steglich, Heavy fermions and quantum phase transitions. Science 329, 1161– 31. K. Penc, J. Solyom,´ Scaling theory of interacting light and heavy fermions in one 1166 (2010). dimension. Phys. Rev. B 41, 704–716 (1990). 8. Q. Si, Quantum criticality and global phase diagram of magnetic heavy fermions. 32. L. Balents, M. P. A. Fisher, Weak-coupling phase diagram of the two-chain Hubbard Phys. Status Solidi 247, 476–484 (2010). model. Phys. Rev. B 53, 12133–12141 (1996). 9. P. Coleman, A. H. Nevidomskyy, Frustration and the Kondo effect in heavy fermion 33. J. C. Xavier, E. Dagotto, Robust d-wave pairing correlations in the Heisenberg Kondo materials. J. Low Temp. Phys. 161, 182–202 (2010). lattice model. Phys. Rev. Lett. 100, 146403 (2008). 10. Y. Zhou, K. Kanoda., T. K. Ng, Quantum spin liquid states. Rev. Mod. Phys. 89, 025003 34. B. S. Chandrasekhar, A note on the maximum critical field of high-field superconduc- (2017). tors. Appl. Phys. Lett. 1, 7–8 (1962). 11. V. Zapf, M. Jaime, C. D. Batista, Bose-Einstein condensation in quantum magnets. Rev. 35. A. M. Clogston, Upper limit for the critical field in hard superconductors. Phys. Rev. Mod. Phys. 86, 563–614 (2014). Lett. 9, 266–267 (1962). 12. M. S. Kim, M. C. Aronson, Heavy fermion compounds on the geometrically frustrated 36. M. Dolfi, B. Bauer, S. Keller, M. Troyer, Pair correlations in doped Hubbard ladders. Shastry–Sutherland lattice. J. Phys. Condens. Matter 23, 164204 (2011). Phys. Rev. B 92, 195139 (2015). 13. K. Hara et al., Quantum spin state in the rare-earth compound YbAl3C3. Phys. Rev. B 37. E. Dagotto, J. Riera, D. Scalapino, Superconductivity in ladders and coupled planes. 85, 144416 (2012). Phys. Rev. B 45, 5744–5747 (1992). 14. Y. Tokiwa, C. Stingl, M. S. Kim, T. Takabatake, P. Gegenwart, Characteristic signa- 38. E. Dagotto, T. M. Rice, Surprises on the way from one- to two-dimensional quantum tures of quantum criticality driven by geometrical frustration. Sci. Adv. 1, e1500001 magnets: The ladder materials. Science 271, 618–623 (1996). (2015). 39. E. Eidelstein, S. Moukouri, A. Schiller, Quantum phase transitions, frustration, and the 15. V. Fritsch et al., Approaching quantum criticality in a partially geometrically frustrated fermi surface in the Kondo lattice model. Phys. Rev. B 84, 014413 (2011). heavy-fermion metal. Phys. Rev. B 89, 054416 (2014). 40. I. Khait, P. Azaria, C. Hubig, U. Schollwock,¨ A. Auerbach, Doped Kondo chain, a heavy 16. T. Giamarchi, C. Ruegg,¨ O. Tchernyshyov, Bose–Einstein condensation in magnetic Luttinger liquid. Proc. Natl. Acad. Sci. U.S.A. 115, 5140–5144 (2018). insulators. Nat. Phys. 4, 198–204 (2008). 41. M. Oshikawa, Topological approach to Luttinger’s theorem and the fermi surface of 17. A. Oosawa, M. Ishii, H. Tanaka, Field-induced three-dimensional magnetic ordering a Kondo lattice. Phys. Rev. Lett. 84, 3370–3373 (2000). in the spin-gap system. J. Phys. Condens. Matter 11, 265 (1999). 42. P. Calabrese, J. Cardy, Entanglement entropy and conformal field theory. J. Phys. 18. T. Nikuni, M. Oshikawa, A. Oosawa, H. Tanaka, Bose-Einstein condensation of dilute Math. Theor. 42, 504005 (2009). magnons in TlCuCl3. Phys. Rev. Lett. 84, 5868–5871 (2000). 43. S. J. Yamamoto, Q. Si, Renormalization group for mixed fermion-boson systems. Phys. 19. S. Miyahara, K. Ueda, Theory of the orthogonal dimer Heisenberg spin model for Rev. B 81, 205106 (2010). SrCu2 (BO3)2. J. Phys. Condens. Matter 15, R327 (2003). 44. A. Ochiai, T. Inukai, T. Matsumura, A. Oyamada, K. Katoh, Spin gap state of s = 1/2 20. S. M. Thomas et al., Hall effect anomaly and low-temperature metamagnetism in the heisenberg antiferromagnet YbAl3C3. J. Phys. Soc. Jpn. 76, 123703 (2007). Kondo compound CeAgBi2. Phys. Rev. B 93, 075149 (2016). 45. T. Matsumura et al., Structural phase transition in the spin gap system YbAl3C3. J. 21. J. Shin, Z. Schlesinger, B. S. Shastry, Kondo-ising and tight-binding models for TmB4. Phys. Soc. Jpn. 77, 103601 (2008). Phys. Rev. B 95, 205140 (2017). 46. K. Hara et al., Quantum spin state in the rare-earth compound YbAl3C3. Phys. Rev. B 22. D. D. Khalyavin et al., Field-induced long-range magnetic order in the spin-singlet 85, 144416 (2012). ground-state system YbAl3C3: Neutron diffraction study. Phys. Rev. B 87, 220406 47. D. S. Fisher, P. C. Hohenberg, Dilute Bose gas in two dimensions. Phys. Rev. B 37, 4936– (2013). 4943 (1988). 23. T. Giamarchi, A. M. Tsvelik, Coupled ladders in a magnetic field. Phys. Rev. B 59, 48. A. J. Millis, Effect of a nonzero temperature on quantum critical points in itinerant 11398–11407 (1999). fermion systems. Phys. Rev. B 48, 7183–7196 (1993).

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