Downloaded by guest on September 27, 2021 ev heavy magnetic emergent the with interactions. Ruderman–Kittel–Kasuya–Yosida associated are the results large-N of numerical the law our between power differences density-dependent The a coupling. large-N by Kondo velocity the in exponential the velocity the fit we to Fermi contrast the In of vector. suppression (con- wave small Fermi the at band) detected duction is quasiparti- gap hybridization Luttinger low-energy A “heavy” cles. of spin manifestation suppressed a highly The is vector. velocity modes wave charge Fermi large and the spin at gapless predic- appear approximation, the N to susceptibilities large the Similar of properties. spin tion low-energy and its competition into charge insights complex com- give vector-dependent numerically describes The wave ordering. Lattice puted magnetic review and Kondo for itinerant (received 2018 1D between 6, April doped Fisk Zachary Member The Board Editorial by accepted and NJ, 2017) Piscataway, 8, University, November Rutgers Coleman, Piers by Edited a liquid Luttinger Khait heavy Ilia a chain, Kondo Doped ooiain hyotie nes-ln prlpae with , www.pnas.org/cgi/doi/10.1073/pnas.1719374115 spiral easy-plane an using obtained model KL They anisotropic the on . 35) 33)]. was (34, works interactions (32, recent RKKY remained in the shown that treating Lut- carefully surface of small Fermi importance the a The of and size controversy the (9) some regarding is surface was there [although Fermi phase parameter interaction large (TLL) tinger 31) a (30, with Liquid hypothesized Tomonaga–Luttinger large-N a the and (29), oscillations Friedel density spin charge of analysis and Matrix the Density From (28). and (DMRG) diagonalization Group exact using 20–27) (11, cally half-filling. theory, from field away and especially Carlo, Monte quantum bosonization, solutions, for coupling Kondo the velocity with Fermi scale renormalized mode) the spin the (i.e., follows: of velocity as spin is the question does crucial the how is its coupling, phase weak change liquid at least Fermi found a at indeed If or (16). (2) correlations ordering and scales magnetic low-energy by metal the HF the in lize order large-N second the (Ruderman– order at at and RKKY coupling emerge Kondo named (14), interactions, Kittel–Kasuya–Yosida) magnetic large- Intersite the physical of the validity for mation the However, Kondo (3). single temperature the of scaling the resembles large-N which Fermi velocity, The coupling) Kondo inverse 13). 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RKKY of effects the large- N the with and associated results is numerical difference The the behavior. law between inter- power the shows on which dependence strength, its action of calculation numerical large-N the the (conduction present by expected small gap consistent hybridization the qualitatively the is twice with gap This at vector. expo- wave gap the Fermi charge electrons’) to low-energy contrast large-N a in the detect by coupling predicted Kondo behavior the the nential of to laws relative power suppressed fits highly is velocity. velocity wave charge spin Fermi the large than and the smaller much at unity, is modes parameter interaction charge Luttinger The and vector. spin gapless by character- a is by ized which described (HTLL), be Liquid can Tomonaga–Luttinger singularities momen- Heavy Their low-energy scales. the velocity measure the and These to tum computing susceptibilities. us by allow spin phase functions and this response probe charge We vector-dependent TLL. wave a of existence vector. the wave Fermi symmetry, large a broken at no are has fermions It massless phases. and these between point The critical vector. wave Fermi small SU the twice at fluctuations collective broken ‡ § oiu e iuds Universit Liquides, des eorique ´ owo orsodnesol eadesd mi:[email protected]. 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PHYSICS KL at weak coupling, J < t, traditionally assumed that RKKY interactions dominate the low-energy physics and inhibit the , which produces heavy . However, A our conclusion is that itineracy and RKKY work hand in hand to produce the HTLL, with a large Fermi wave vector, which bears resemblance to HF states in higher dimensions. Model B We study the SU(2) 1D KL model:

X † X ~ H = −t ci,s ci+1,s + H.c + J Si ·~si . [1] i,s i

J is the Kondo interaction. cis annihilates an electron at posi- α 1 P † α α tion i = 1 ... L with z spin s; si = 2 ss0 cis σss0 cis0 , S are spin-half operators of the conduction electrons and localized f spins, and σα, α = x, y, z are Pauli matrices. The conduction Fig. 2. Charge susceptibility vs. wave vector for nc = 0.875, J/t = 2.5. (A) electron number Nc = nc L defines the “small” Fermi wave vector ? π large-N approximation. (B) DMRG calculation. kF denotes the large Fermi kF = 2 nc . The “large” (Luttinger theorem) Fermi wave vector is surface wave vector. Finite-field scaling of DMRG reveals divergent peaks at ? π k = kF + . ? ? F 2 2kF and 4kF as expected for a TLL. B, Inset shows a nondivergent peak at The phase diagram in the nc , J /t plane is shown in Fig. 1. twice the small Fermi wave vector 2kF , which is attributed to the inverse of Here, we avoid the ferromagnetic regions (FMs) FM1 and FM2 the hybridization gap 2r0 depicted in Fig. 6. arb, arbitrary units. and choose our parameters to be well within the weak-coupling paramagnetic regime. Note that, according to the large-N theory 0 < J  4π sin(k ? ) z z ? −K (Eq. 11), weak coupling is defined by t F . h(Si + si )i = B1 cos(2kF x) xi [4] Luttinger Parameters and Susceptibilities and

The uniform spin and charge susceptibilities are given by dif- K +1 X † ? − 2 ferentiating the ground-state energy E0(L, Nc , M ), where M = hcis cis i = A1 cos(2kF xi ) xi P z z i (Si + si ) is the conserved magnetization: s ? −2K +A2 cos(4kF xi ) xi , [5]  2 −1  2 −1 1 ∂ E0 1 ∂ E0 χ (L) = , χ (L) = . [2] ? s L ∂M 2 c L ∂N 2 which allows us, in principle, to extract kF and K . Other wave c vectors are associated with smaller-amplitude oscillations, which By the TLL theory, these are related to the spin and charge are hard to extract from Eqs. 4 and 5. Therefore, we use a com- velocities by plementary approach and calculate the susceptibilities by adding wave vector-dependent source terms to the Hamiltonian:

−1 −1 0 z z vs = χs /(2π), vc = χc K /(2π), [3] H = −hq (Sq + sq ) − µq ρq ,

z z where Sq , sq , and ρq are the lattice cosine transforms of the z z respectively, where K is the Luttinger interaction parameter. In operators Si , si , and ρi . The susceptibilities are obtained by a TLL theory, the spin and Friedel oscillations differentiating the DMRG ground-state energies:

near the boundaries are given to leading order by (20, 36) 2 2 1 ∂ E0 1 ∂ E0 χs (q) = − 2 , χc (q) = − 2 . [6] L ∂hq L ∂µq

To avoid finite lattice effects, we take hq and µq to be larger than the respective finite size gaps. Methods We use open boundaries with U(1)¶ (unitary group of degree 1) and SU(2)# DMRG (28). Lattice sizes were L ≤ 192. We retain up to 5,500 states in the reduced density matrix; 28 sweeps were sufficient for good convergence. The DMRG relative truncation error was less than 10−8. Friedel spin density oscillations were found at twice the large Fermi wave ? vector 2kF = 2kF + π, in agreement with Luttinger’s theorem. Due to the ? very low values of K, the signature of 2kF in the charge Friedel oscillations is too weak for detection (11, 23–27) (SI Appendix, Fig. S5). The Luttinger parameter K was determined by measuring the power law ? ? singularities of the density operator nq at 2kF and 4kF . TLL theory yields the scaling relation (SI Appendix has a complete derivation):

∆(q) 1 ∂E0 2−∆(q) nq = − ∼ µq , [7] Fig. 1. Schematic phase diagram of the KL model in one dimension; nc is L ∂µq the conduction electron density, and J/t is the dimensionless Kondo cou- pling constant. FM1 (11) and FM2 (20–22) are shown. The paramagnetic phase (blue) is characterized as an HTLL. The DMRG calculations reported ¶ Calculations were performed using the ITENSOR library. here are carried out at densities nc = 0.6, 0.67, and 0.875 marked with thick # vertical green lines. Calculations were performed using the SYTEN library.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1719374115 Khait et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 aevector wave calculations. DMRG our of approached limitations which numerical of entanglement, the values ground-state lower of increase The rapid 1). Fig. in (shown constant coupling of ranges fillings, three chose We Results Eq. in of values n where niaeefcso togrplieinteractions. repulsive strong of of effects values indicate The 1. Fig. 4. Fig. the by at peak divergent large-N 3. Fig. hi tal. et Khait func- a as decrease monotonic a note of we tion depicted; are detec- the note We respectively. at 3, peaks and divergent of 2 tion Figs. in depicted are (36). oscillations 2k hc rl ofimteTLpaewt ag omnFermi common 4k large a with vector phase wave TLL the confirm firmly which q F F h ree siltoso h pndniyaedmntdby dominated are density spin the of oscillations Friedel The nFg ,tevle fteLtigrparameter Luttinger the of values the 4, Fig. In susceptibilities spin and charge vector-dependent wave The ? ? ttetowv etr 2k vectors wave two the at sepce o ml ausof values small for expected as siltoswr o ekfrdtcini h Friedel the in detection for weak too were oscillations 5. ∆(q B A MGcluain iiefil cln eel a reveals scaling Finite-field calculation. DMRG (B) approximation. utne neato parameter interaction Luttinger pnssetblt s aevco for vector wave vs. susceptibility Spin 4k K J xrce from extracted = /t F ? 2k inln togrrplieitrcin tweak at interactions repulsive stronger signaling , 2k k em,since terms, F ? F ? ) F ? h hresco hw rnucdpa at peak pronounced a shows sector charge The . q = h hredniyoclain r dominated are oscillations density charge The . = K +1 2 2k K F . ? and r,abtayunits. arbitrary arb, . ∆ 3truhu h TLpae(ntetext) the (in phase HTLL the throughout 0.33 n n rmtecag est ree oscillations Friedel density charge the from and c ∆(q F ? K .,06,ad085 ihassociated with 0.875, and 0.67, 0.6, = 2k n 4k and J < = F ? /t 4k 0.33 nbt pnadcag sectors, charge and spin both in ihnteprmgei region paramagnetic the within F ? F ? ) efidgo gemn o the for agreement good find We . = K K (Eq. 2K ln h he ie hw in shown lines three the along . r h cln iesosof dimensions scaling the are J /t .Tesubdominant The 5). n c eelmtdb the by limited were = 0.875, J K /t (n = c .(A) 2.5. , J /t ) opigi gemn iherircluain yShibata by Eq. calculations earlier with (29). al. agreement et in coupling 7). (Fig. susceptibility of dependence weaker a Note ceptibility. of decrease a laws power are 5. Fig. prxmto ilsa fetv yrdzdbn structure, band 6: hybridized Fig. in effective depicted is constraint which an a yields to approximation subject are which Eq. in spins localized The Large-N other by shared is well. gap rele- as hybridization is fillings the HTLL of the larger appearance which The much for vant. scales scale, energy high-energy characteristic gap the the hybridization than is inverse this the Clearly, to attributed (Discussion). is which vector, wave respective velocities with moderately fit as can significantly We phase. HTLL law of limit hallmark the weak-coupling is ness” the in suppressed B A ecmuetesi n hrevlcte fteTLby TLL the of velocities charge and spin the compute We i.2B Fig. eseta h hressetblt aisonly varies susceptibility charge the that see we 5B, Fig. In h pnvelocity spin The 3. (J pnvlct safnto of function a as velocity Spin (A) /t Approximation ) v α hw odvrigpa ttietesalFermi small the twice at peak nondiverging a shows v c ( s A n . r tlata re fmgiuelre hnthe than larger magnitude of order an least at are c v as ) s n where , ∼ n c c A(n J prahsunity. approaches nfr ( q Uniform (B) (Discussion). increases /t c S )(J i γ and /t = 1. v ) α(n s 1 2 1 (J n < α < r ersne ysaefermions: slave by represented are c X c s ) ent infiatices of increase significant a note We . ,s o u aaees h charge the parameters, our For . /t 0 shighly is 5A, Fig. in shown as ), f i † ,s h power The 8.3 J σ χ P /t ss γ c on 0 tdfeetfilns oi lines fillings. different at s f i f NSLts Articles Latest PNAS ,s i n † J ,s c 0 , f /t and i ,s  1 = J /t hs“heavi- This 1. h large-N The . hnfrtespin the for than v = s α )cag sus- charge 0) oapower a to increases | α f5 of 3 and [8]

PHYSICS ii) In the large-N theory, the f fermions cannot contribute to the charge susceptibility. χc is completely determined by the conduction electrons susceptibility and is weakly dependent on the Kondo coupling. The DMRG results for χc are also weakly dependent on J /t, as shown in Fig. 5. iii) The hybridized bands in the large-N theory predict a minimal hybridization gap at the small Fermi wave vector kF (Fig. 6). This gap should be proportional to the square root of the spin velocity. The DMRG indeed finds a small peak at 2kF (Fig. 2B, Inset). This peak can be interpreted as a signature of a hybridization gap at ±kF , albeit with a different parametric dependence on J /t than given by Eq. 12. This qualitative feature is in support of a hybridized band description of the HTLL. Differences between the large N and the DMRG results are

Fig. 6. Band structure of the large-N Hamiltonian. k is the tight bind- especially noteworthy. As seen in Fig. 7, the variation of the ± ing kinetic energy, Ek are the hybridized bands, 2r0 is the hybridization spin velocity with the Kondo coupling is well-fit by power laws. gap. The shaded area denotes the occupied part of the Fermi surface. The large-N predictions cannot be reconciled with the numer- Blue-colored f and orange-colored c denote the f-character and c-character ical data unless large renormalization factors are introduced parts of the band, respectively. Note the suppression of the spin velocity vs into the exponent of Eq. 12. While it is hard to distinguish compared with the charge velocity given by vc. α between a power vs ∝ (J /t) and a general exponential function e−F(nc )t/J , it requires unjustifiably large renormalization factors Large−N X † † † F (nc ), especially at lower fillings. In addition, we find that the H =  c c + r c f + H.c +  f f k ks ks 0 ks ks f ks ks hybridization gap scales weakly with the Kondo coupling com- ks pared with the high-power law suppression of the spin velocity X ± † 12 = Ek α±,k,s α±,k,s . [9] (Fig. 5A). This differs from the large-N prediction (Eq. ). ks Fig. 5A shows that the DMRG-obtained power laws α increase toward half-filling, nc → 1. The reason for the increase in α(nc ) The bare conduction electron band structure is k = −2t cos(k), is still uncertain, but it may be associated with the approaching ± and the large-N hybridized bands Ek are into the Kondo at nc = 1, where both spin and charge modes are gapped (11). r  +    −  2 E ± = k f ± k f + r 2. [10] k 2 2 0 Summary Our DMRG results of the spin and charge wave vector- f and r0 are variational parameters, which depend on J /t and dependent susceptibilities establish the HTLL phase of the N = nc . Solving the mean field equations at weak coupling yields (37) 2 KL model at weak coupling and away from half-filling. In addi- ! tion, we find numerical evidence for hybridized band features ? k − kF 1 2 F ρ J revealed by a nondivergent peak of the wave vector-dependent r0 = e 0 , [11] 2ρ0 ? charge susceptibility at precisely twice the small Fermi wave vec- kF tor 2kF . Altogether, our results strongly support the fact that, at 1 where ρ0 = ? is the (single-spin) conduction electron’s least in one dimension, the RKKY interactions do not destroy 4πt sin(kF ) ? 2 the heavy quasiparticles at kF but “work with them” to produce at the large Fermi wave vector. r0 determines ? 2 ? the HTLL. At this point, it is worth commenting on recent works the Fermi velocity suppression as vF = 2tr0 sin(kF ), and r0 yields (34, 35) studying the anisotropic KL, where it was found that, the minimal hybridization gap at kF . As shown in Fig. 6, the small in the presence of a strong easy-plane anisotropy (XXZ type), Fermi wave vector kF marks a sharp cross-over in the character of the quasiparticles from c to f fermions. Discussion The large-N theory effectively describes noninteracting fermions (in essence, a TLL with K = 1). Its spin and charge wave vector- dependent susceptibilities are compared with the DMRG in Figs. 2 and 3. Large-N theory exhibits a weak logarithmic singularity at ? 2kF , while the DMRG exhibits a K -dependent power law diver- ? gence. The DMRG peak in χc (q) at 4kF , absent in the large-N theory, is also associated with K < 1. Similarities between large-N theory and the DMRG are as follows. i) The ratio of charge to spin susceptibilities. In the large-N theory,

χ 1 s −2 ρ J = r0 ∝ e 0 , [12] χc

which follows from the dominance of the f fermions charac- Fig. 7. Uniform spin susceptibility, which is related to the velocity of Fig. 5 ? χs via χ = 1/(2πv ). Solid straight lines are a fits to power laws χ (J/t)−α. ter at kF . The DMRG also obtains  1 throughout the s s s ∝ χc 1 Large−N ρ J studied parameter regime. Dashed lines are fits to the large-N theory χs ∝ e 0 .

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The et P upt Aynajian in (1997) 13. quasiparticles Heavy-fermion (1988) K GG Lonzarich L, Ueda Taillefer 12. M, Fermi Sigrist the and H, theorem Luttinger’s Tsunetsugu to 11. approach Topological (2000) M Oshikawa 10. hi tal. et Khait od opigcnsrea sflgiefrftr work. future for guide useful a as serve can coupling Kondo gap. competi- hybridization the the decoupled of in the formation of role formation important the an between play tion vectors wave RKKY quarter- the At observed. was the gap filling, charge a (41)], quarter-filling for and by, gap hybridization the 40). 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Schollw C, renormal- Kollath matrix AJ, density the Daley using 40. evolution Real-time (2004) AE Feiguin correlations SR, spin White spiral incommensurate 39. and Dimerization (1996) I Affleck heavy SR, White of properties 38. universal and lattice Anderson The (1987) K waves Levin density A, charge Auerbach and oscillations 37. Friedel (2002) DJ Scalapino I, Affleck SR, White 36. 5 cimlD,TvlkA,YvuhnoO 21)Lweeg rpriso the of properties energy Low (2016) ideal OM protected Yevtushenko and AM, transition Tsvelik DH, phase Schimmel Quantum 35. (2015) O Yevtushenko one-dimensional A, the Tsvelik in 34. exponent surface Correlation Fermi (2004) E Small Miranda JC, (2002) Xavier E 33. Miranda E, Novais J, Xavier 32. (2004) T Giamarchi a 31. I. as fluids. quantum model one-dimensional lattice of theory’ Kondo liquid One-dimensional ‘Luttinger (1981) (1997) FDM Haldane K Ueda 30. A, groups. Tsvelik renormalization N, quantum Shibata for formulation 29. matrix the by Density studied (1992) model SR lattice White Kondo one-dimensional 28. model. The lattice (1999) Kondo K the Ueda of N, diagram Shibata Phase (1993) 27. M Sigrist H, Tsunetsugu K, Ueda 26. the of state ground W Spin-liquid M, (1992) Troyer M Sigrist 25. K, Ueda Y, Hatsugai H, Tsunetsugu Kondo 24. strong-coupling the one-dimensional in (1992) the TM Rice K, in Ueda H, Tsunetsugu state M, Sigrist Ferromagnetic 23. (2012) N Kawakami R, Peters 22. ela h al nttt o hoeia hsc ne ainlScience National under Physics Theoretical hospitality. its for for PHY-1125915 Institute as NSF 1111/16 Grant Kavli Grant Foundation the Foundation Ini- U.S. Science Science as Nanosystems Israel Binational well and the and (ExQM) States–Israel 2016168 and C.H. United Grant Matter Network acknowledges Foundation hospitality. Elite A.A. Quantum Bavarian kind Munich. Physics Exploring the tiative their the of the School thanks for through Graduate P.A. Technion support. funding thank the technical acknowledge also extensive We of his discussions. Department for beneficial for Cohen Podolsky David Daniel and Kesselman, ACKNOWLEDGMENTS. they the as in regime, dimension. one visible heavy-mass in weak-coupling, be are the should in gap) sector small hybridization gapped charge the (i.e., of signatures surface particular, Fermi In HFs. systems and atom KL sional cold and experi- issues. wires these future on in that light shed KL hope also the We may of theory. realizations to challenge mental and a dimerization as between remains interplay the phase: HTLL out the velocity in emerge spin dif- to the parametrically expected are two scales approach, energy ferent analytical any in Specifically, esn ere eecudb eeatt h high-dimen- the to relevant be could here learned Lessons interaction. Exp Theor Mech Stat J 2004:P04005. spaces. hilbert effective adaptive using renormalization-group group. ization lattice. Kondo the to Analogies chain: spin zigzag the 495–499. in pp Boston), (Springer, SK Malik LC, Gupta eds Fermions, Heavy systems. fermion ladders. and chains in od hi nterk regime. rkky the in chain Kondo chain. Kondo a in transport model. lattice model. lattice Kondo 121. Vol interacting 1D general the to extension their . Fermi and spinless model Luttinger the of Properties liquid. Tomonaga-Luttinger Lett Rev Phys method. group renormalization matrix density Matter Condens B Phys study. Carlo Monte quantum A model: dimension. one in lattice Kondo half-filled model. Kondo-lattice one-dimensional the of regime model. lattice hsRvLett Rev Phys rzD(93 ermgeimo h n-iesoa Ksondo-lattice one-dimensional the of Ferromagnetism (1993) D urtz ¨ 69:2863–2866. hsRvB70:075110. Rev Phys B Rev Phys hsRvLett Rev Phys hoeia n xeietlApcso aec lcutosand Fluctuations Valence of Aspects Experimental and Theoretical hsCSldSaePhys State Solid C Phys J unu hsc nOeDimension One in Physics Quantum v hsRvB Rev Phys hsRvB Rev Phys s 186:358–361. etakEe eg yvi apn,Anna Capponi, Sylvain Berg, Erez thank We n h yrdzto a at gap hybridization the and 86:165107. 90:247204. hsRvLett Rev Phys hsRvB Rev Phys 93:076401. c ,VdlG(04 iedpnetdensity-matrix Time-dependent (2004) G Vidal U, ock ¨ 65:214406. e Phys J New 65:165122. hsRvB Rev Phys 56:330–334. 115:216402. hsRvB Rev Phys 14:2585–2609. 18:053004. hsCnesMatter Condens Phys J 47:2886–2889. K ρ NSLts Articles Latest PNAS hsRvB Rev Phys fteoedmninlKondo one-dimensional the of 46:3175–3178. Ofr nvPes Oxford), Press, Univ (Oxford hsRvB Rev Phys 2k F 46:13838–13846. hybridization 2k 54:9862–9869. F 11:4289–4290. Sorting . | f5 of 5

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