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Frustration-Driven Multi-Magnon Condensates Frustration-driven multi-magnon condensates Credit: Francis Pratt / ISIS / STFC Oleg Starykh University of Utah ! RECENT PROGRESS IN LOW-DIMENSIONAL QUANTUM MAGNETISM LDQM2016 5-16 September 2016, EPFL, Lausanne Why do we do it? Immense potential for practical applications Collaborators Leon Balents, KITP, UCSB Andrey Chubukov, FTPI, U Minnesota Jason Alicea, Caltech General motivation: Exotic but ordered phases ordered ! spin nematic! quantum! phases hidden order spin liquids spin nematic composite order parameter Outline • Experimental evidence: Frustrated ferromagnets and magnon binding • Basic theory and some numerics • Field theory of the Lifshitz point magnon “BCS” superconductor • Spin-current state near the end-point of 1/3 magnetization plateau exotic superconductorof magnons RAPID COMMUNICATIONS PHYSICAL REVIEW B 80, 140402͑R͒͑2009͒ Emergent multipolar spin correlations in a fluctuating spiral: The frustrated ferromagnetic 1 spin-2 Heisenberg chain in a magnetic field Julien Sudan,1 Andreas Lüscher,1 and Andreas M. Läuchli2,* 1Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), CH-1015 Lausanne, Switzerland 2Max Planck Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany ͑Received 22 July 2008; published 7 October 2009͒ 1 We present the phase diagram of the frustrated ferromagnetic S= 2 Heisenberg J1 −J2 chain in a magnetic field, obtained by large scale exact diagonalizations and density matrix renormalization group simulations. A vector chirally ordered state, metamagnetic behavior and a sequence of spin-multipolar Luttinger liquid phases up to hexadecupolar kind are found. We provide numerical evidence for a locking mechanism, which can drive spiral states toward spin-multipolar phases, such as quadrupolar or octupolar phases. Our results also shed light 1 on previously discovered spin-multipolar phases in two-dimensional S= 2 quantum magnets in a magnetic field. DOI: 10.1103/PhysRevB.80.140402 PACS number͑s͒: 75.10.Jm, 75.30.Kz, 75.40.Cx, 75.40.Mg Spiral or helical ground states are an old and well- ploy exact diagonalizations ͑EDs͒ on systems sizes up to understood concept in classical magnetism,1 and several ma- L=64 sites complemented by density matrix renormalization terials are successfully described based on theories of spiral group ͑DMRG͒͑Ref. 11͒ simulations on open systems of states. For low spin and dimensionality however quantum maximal length L=384, retaining up to 800 basis states. fluctuations become important and might destabilize the spi- The classical ground state of Hamiltonian ͑1͒ is ferromag- ral states. Given that spiral states generally arise due to com- netic for J2 Ͻ1/4 and a spiral with pitch angle ␸ peting interactions, fluctuations are expected to be particu- =arccos͑1/4J2͒෈͓0,␲/2͔ otherwise. The Lifshitz point is larly strong. located at J2 =1/4. In a magnetic field the spins develop a A prominent instability of spiral states is their intrinsic uniform component along the field, while the pitch angle in 2 twist ͗Si ϫSj͑͘vector chirality͒. It has been recognized that the plane transverse to the field axis is unaltered by the field. finite temperature3 or quantum4 fluctuations can disorder the The zero field quantum mechanical phase diagram for 1 spin moment ͗Si͘ of the spiral, while the twist remains finite. S= 2 is still unsettled. Field theoretical work predicts a finite, 5 12,13 Such a state is called p-type spin nematic. In the context of but tiny gap accompanied by dimerization for J2 Ͼ1/4, quantum fluctuations such a scenario has been confirmed re- which present numerical approaches are unable to resolve. 6 cently in a ring-exchange model, while possible experimen- The classical Lifshitz point J2 =1/4 is not renormalized for tal evidence for the thermal scenario has been presented in.7 1 S= 2 , and the transition point manifests itself on finite sys- The twist also gained attention in multiferroics, since it tems as a level crossing between the ferromagnetic multiplet couples directly to the ferroelectricity.8 and an exactly known singlet state.14 The theoretical phase In this Rapid Communication we provide evidence for the diagram at finite field has recently received considerable existence of yet a different instability of spiral states toward attention,15–17 triggered by experiments on quasi one- spin-multipolar phases. The basic idea is that many spin- dimensional cuprate helimagnets.18–20 One of the most pecu- multipolar order parameters are finite in the magnetically liar features of the finite size magnetization process is the NaCuMoO4 OH as a Candidate Frustrated J1–J2 Chain Quantum Magnet ( ) ordered spiral state, but that under a suitable amount of fluc- appearance of elementary magnetization steps of Kazuhiro Nawa1,∗ Yoshihiko Okamoto1,† Akiratuations Matsuo1, Koichi the Kindo primary1, Yoko Kitahara spin2, Syota order Yoshida2 is, lost, while a spin- z Shohei Ikeda2, Shigeo Hara3, Takahiro Sakurai3,SusumuOkubo4, Hitoshi Ohta4, and Zenji Hiroi1 ⌬S =2,3,4 in certain J2 and m regions. This has been attrib- 1Institute for Solid State Physics, Themultipolar University of Tokyo, orderKashiwa, parameter Chiba 277-8581, Japan survives. We demonstrate this 2Graduate School of Science, Kobe University, Nada, Kobe 657-8501, Japan uted to the formation of bound states of spin flips, leading to 3Center for Supports to Research and Educationmechanism Activities, Kobe University, based Nada, on the Kobe 657-8501, magnetic Japan field phase diagram of a 4MolecularNew Photoscience Research system: Center, Kobe University, Nada, Frustrated Kobe 657-8501, Japan dominant spin-multipolar correlations close to saturation. A 1 (Dated: September 5, 2014) prototypical model, the frustrated S= 2 Heisenberg chain detailed phase diagram is however still lacking. In a frustrated J1–J2 chain with the nearest-neighbor ferromagnetic interaction J1 and the next- nearest-neighbor antiferromagnetic interactionwithJ2,novelmagneticstatessuchasaspin-nematicstate ferromagnetic nearest-neighbor and antiferromagnetic We present our numerical phase diagram in the J2 /͉J1͉ vs. are theoretically expected. However,ferromagnet theynext have been nearest-neighbor rarely examined in experiments interactions. because of the Furthermore we show difficulty in obtaining suitable model compounds. We show herethatthequasi-one-dimensional m/msat plane in Fig. 1. At least five different phases are antiferromagnet NaCuMoO4(OH), which comprises edge-sharing CuO2 chains, is a good candidate J1–J2 chain1d antiferromagnet. S=1/2 The exchangechainthat interactions this instability are estimated as providesJ1 = −51 K a and naturalJ2 and unified understand- present. The low magnetization region consists of a single =36Kbycomparingthemagneticsusceptibility,heatcapacity, and magnetization data with the data obtained using calculations by theing exact diagonalizat of previouslyion method. High-field discovered magnetization two-dimensional spin- vector chiral phase ͑gray͒. Below the saturation magnetiza- measurements at 1.3 K show a saturation above 26 T with little evidence9, of10 a spin nematic state expected just below the saturation field, whichmultipolar is probably due phases. to smearing effectsJ1 caused<0 by thermalFM tion we confirm the presence of three different multipolar fluctuations and the polycrystalline nature of theTo sample. be specific, we determine numerically the phase dia- Luttinger liquid phases ͑red, green, and blue͒. The red phase 16 Low-dimensional quantum spin systems with geomet-gram of the following Hamiltonian: extends up to J2 ϱ, and its lower border approaches TABLE I. Candidate compounds for the J1–J2 chain system. rical frustration and/or competing magnetic interactions + → Listed are the nearest-neighbor intrachain interaction J1,the m=0 in that limit. All three multipolar liquids present a have attracted much attention in the field of magnetism. z J2>0 AF next-nearest-neighbor interaction J2,thebondanglesofCu- Low dimensionality, quantum fluctuations, and frustra- H = J1 Si · Si+1 + J2 Si · Si+2 − h Si , ͑1͒ O-Cu paths for J1͚,theantiferromagnetictransitiontemper-͚ ͚ crossover as a function of m/msat, where the dominant cor- tion are three ingredients that may effectively suppress ature at zero field TiN,andthesaturationfieldiHs. i conventional magnetic order and lead us to unconven- relations change from spin-multipolar close to saturation to tional magnetic order or exotic ground states such as a Compound J1, J2 ∠ Cu-O-Cu TN Hs quantum spin liquid[1, 2]. and we set J1 =−1, J2(K)Ն0 in(deg) the (K) following. (T) Si are spin-1/2 spin-density wave ͑SDW͒ character at lower magnetization. AfrustratedJ1–J2 chain of spin 1/2 defined as Li2ZrCuO4[12, 13] −151, 35 94.1 6.4 - operatorsRb2Cu at2Mo3 siteO12[14,i 15],− while138, 51 89.9,h 101.8denotes< 214 the uniform magnetic One also expects a tiny incommensurate p=2 phase close to z H = J1 ! sl · sl+1 + J2 ! sl · sl+2 − h ! sl (1) 91.9, 101.1 z 17 field. The magnetization− is defined as m 1 L S . We em- the p=3 phase, which we did not aim to localize in this l l l PbCuSO4(OH)2[16–18] 100, 36 91.2, 94.3 2.8 5.4 ª / ͚i i LiCuSbO4[19] −75, 34 89.8, 95.0 < 0.1 12 provides us with an interesting example: the competi- 92.0, 96.8 tion between the nearest-neighbor (NN) ferromagnetic LiCu2O2[20–22] −69, 43 92.2, 92.5 22.3 110 interaction J1 and the next-nearest-neighbor (NNN)1098-0121/2009/80 an- LiCuVO4[23–31]͑14͒−/14040219, 44 95.0͑4͒ 2.1 44.4 140402-1 ©2009 The American Physical Society − tiferromagnetic interaction J2 causes various quantum NaCuMoO4(OH) 51, 36 92.0, 103.6 0.59 26 states in magnetic fields h[3–7]. Realized in low fields is a long-range order of vector chirality defined as (sl × s l+n)z (n =1, 2).β As the fieldTeVO increases,4 spin correlations K. Nawa et al, arXiv:1409.1310 change markedly because− bound magnon pairs are stabi- such as Li2ZrCuO4[12, 13], Rb2Cu2Mo3O12[14, 15], lizedPregelj by ferromagnetic et al., Nat.Comm.2015J1.
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