Classical and quantum anisotropic Heisenberg antiferromagnets

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1 Classical and quantum anisotropic Heisenberg antiferromagnets W. Selke 1, G. Bannasch 1,2, M. Holtschneider 1, R. Leidl 1, I.P.McCulloch 3, D. Peters 1, and S. Wessel 4,1 (1) Institut für Theoretische Physik und JARA-SIM, RWTH Aachen, Germany (2) MPI für Physik komplexer Systeme, Dresden, Germany (3) University of Queensland, Brisbane, QLD 4072, Australia (4) Institut für Theoretische Physik III, Universität Stuttgart, Germany In part funded by: Excellence initiative of the German federal and state governments

2 Outline XXZ antiferromagnets on square and cubic lattices Classical XXZ models S=1/2 XXZ antiferromagnet in two dimensions Adding single ion crystal field anisotropies Classical antiferromagnets in two and three dimensions S=1 anisotropic Heisenberg spin chain at T=0 Conclusions

3 Classical XXZ Heisenberg antiferromagnet Generic and fundamental model for spin-flop structures multicritical (bicritical, tetracritical) points H XXZ =J i,j [ (Si xsx j +Sy i Sy j )+Sz i Sz j ] H i Sz i Mean field theory, renormalization group, Monte Carlo simulations, numerous related experiments H spin-flop (SF) paramagnetic spin-flop antiferromagnetic (AF) antiferromagnetic T

4 Classical XXZ antiferromagnet on a square lattice spin-flop paramagnetic spin-flop to paramagnetic phase boundary: Kosterlitz-Thouless phase transition antiferromagnetic k B T/J Obtained using Monte Carlo simulations, = 0.8; Holtschneider,WS+Leidl, PRB (2005), in agreement with Landau+Binder, PRB(1981) antiferromagnetic to paramagnetic phase boundary: second order phase transition of Ising type

5 Boundary of the antiferromagnetic phase paramagnetic z 2 <(M st ) > spin-flop antiferromagnetic k B T/J (T c -T) 1/4 L = 10 L = 20 L = 40 L = T c -T / T c order parameter of the AF phase: (Mst) z 2 ) M z st = 1 L 2 ( i A S z i i B S z i = 1.0 k B T c/j = ± critical behavior of Ising universality class

6 Boundary of the spin-flop phase L = 10 L = 20 L = 40 L = spin-flop paramagnetic xy 2 <(M st ) > antiferromagnetic k B T/J T KT k B T/J order parameter of the SF phase (M xy st ) 2 = (M x st) 2 + (M y st) 2 L η = 2.7 k B T c/j = ± η < 1 4 T < T KT = 1 4 T = T KT = 2 T > T KT

7 Boundary of the spin-flop phase L 1 =80, L 2 =60 L 1 =60, L 2 =40 L 1 =40, L 2 =30 L 1 =30, L 2 =20 L 1 =20, L 2 =10 η eff η eff = 1/ T KT k B T/J effective exponent η eff (L) = lim η eff(l) η L d ln (M xy st ) 2 L d ln L < 1 4 T < T KT = 1 4 T = T KT = 2 T > T KT critical behavior consistent with Kosterlitz-Thouless transition

8 Antiferromagnetic to spin-flop transition spin-flop antiferromagnetic k B T/J down to low temperatures: antiferromagnetic phase boundary: second order (Ising) spin-flop phase boundary: Kosterlitz-Thouless transition narrow intervening disordered phase Holtschneider,WS+Leidl, PRB(2005) 3500 e.g. staggered susceptibility: χ = ( (M L2 st) z 2 Mst z 2) k BT χ L = 40 L = 80 L = 120 L = 160 finite-size behaviour: χ max L γ γ = 7 4 Ising k B T/J

9 Antiferromagnetic to spin-flop transition spin-flop antiferromagnetic k B T/J down to low temperatures: antiferromagnetic phase boundary: second order (Ising) spin-flop phase boundary: Kosterlitz-Thouless transition narrow intervening disordered phase Holtschneider,WS+Leidl, PRB(2005) e.g. staggered susceptibility: χ = ( (M L2 st) z 2 Mst z 2) k BT finite-size behaviour: χ max L γ γ = 7 4 Ising χ max L 7/ L

10 Antiferromagnetic to spin-flop transition spin-flop antiferromagnetic k B T/J down to low temperatures: antiferromagnetic phase boundary: second order (Ising) spin-flop phase boundary: Kosterlitz-Thouless transition narrow intervening disordered phase Holtschneider,WS+Leidl, PRB(2005) subsequent analysis suggests: hidden zero temperature bicritical point with phase boundaries meeting at T = 0 Zhou,Landau+Schulthess, Phys. Rev.B(2006)

11 New aspect: role of biconical (BC) structures θ θ 1 2 Ground state analysis: highly degenerate groundstate at field separating AF and SF structures, with tilt angles, Θ 1=A, Θ 2=B, at neighboring sites interrelated by: ( ) Θ B =arccos cos Θ A 1 2 cos Θ A Simulations: BC structures persist at low T, in transition region between AF and SF phases, leading, presumably, to the narrow disordered phase in between the AF and SF phases ( hidden TETRAcritical point at T=0) Holtschneider,Wessel+WS, PRB(2007) Holtschneider+WS, PRB(2007);EPJ B(2008)

12 Simulations: BC structures at low temperatures spin-flop antiferromagnetic k B T/J Part of phase diagram of XXZ model on square lattice Biconical fluctuations: Probability p 2 to have tilt angles Θ A and Θ B for neighboring spins at k B T/J = 0.255, = p 2 is proportional to gray scale. Solid line: Θ A = Θ A (Θ B ) at T =0

13 Classical XXZ model on cubic lattice paramagnetic SF para 8 6 SF AF AF k B T/J Simulated phase diagram of the XXZ antiferromagnet, = 0.8, with, inset, the vicinity of the bicritical, Heisenberg point, at which the AF, SF, and the disordered phases meet; k B T t /J = ± 0.015; H t /J = 3.90 ± 0.03 Bannasch+WS,EPJ B(2009),WS, PRE(2011); agrees with Landau+Binder, PRB(1978); confirmed by Hu,Tsai +D.P.Landau, PRE(2014); Freire+Plascak, PRE(2015); related: Calabrese et al. PRB(2003); Folk et al.,pre (2008)

14 Biconical fluctuations at low T Solid line denotes interrelation of tilt angles at T = 0 and field separating AF and SF phases (as for d=2): ( ) 1 Θ B =arccos 2 cos Θ A cos Θ A Probability p 2 of finding tilt angles Θ A and Θ B at neighboring sites, at k B T/J = 0.7 in the transition region between the AF and SF phases, showing (i) coexistence of AF and SF structures signaling a first order transition (ii) the relevance of interrelated biconical fluctuations arising from the ground state at the field separating AF and SF phases Bannasch+WS, EPJ B (2009)

15 Quantum spin- 1 2 XXZ model on square lattice KT spin-flop paramagnetic Ising 0.5 antiferromagnetic k B T/J = 2/3. Phase diagram obtained from Q-MC simulations (SSE): Holtschneider, Wessel+WS,PRB(2007) H = J i,j [ ] (Ŝx i Ŝx j + Ŝy i Ŝy j ) + Ŝz i Ŝz j H i Ŝ z i

16 Quantum spin- 1 2 XXZ model on square lattice KT spin-flop paramagnetic spin-flop paramagnetic 1.0 Ising CE TP 0.5 antiferromagnetic 1.20 antiferromagnetic k B T/J k B T/J Schmid,Todo,Troyer+Dorneich, PRL(2002): tricritical point (TP), critical endpoint (CE) direct (first order) antiferromagnetic spin-flop transition Check: jump in magnetization?

17 Magnetization histograms 25 L = 32, = L = 150, = p(<m z >) ˆM z := 1 L 2 i Ŝ z i histogram displays two-peak structure <M z > k B T/J = 0.13 coexistence of the two phases defined by equal weight of the peaks

18 Magnetization histograms (M z ) max paramagnetic Comparison to previous results: larger lattices suggest single peak in the thermodynamic limit 0.04 Schmid et al. antiferromagnetic /L Peak-positions at coexistence; Holtschneider,Wessel+WS, PRB(2007) compared to Schmid et al., PRL 2002 tricritical point has to be shifted to lower temperatures, but AF-SF transition of first order at low T plausible, because no simulational evidence for biconical structures

19 Classical anisotropic XY model on square lattice H XY = J [ S x i Sj x + S y ] i Sy j H i,j i S x i SF PM θ SF θ SF θ A θ B 5 4 spin-flop AF k B T/J (b) (c) 3 2 H c1 paramagnetic (a) θ SF (d) θ A (e) θ B 1 antiferromagnetic (a)antiferro (d)spin-flop (e)bidirectional k B T/J = 0.8 Holtschneider+WS, PRB(2007); EPJ B(2008)

20 Role of bidirectional fluctuations At T = 0, degenerate bidirectional structures occur at the field separating the AF and SF structures, with the tilt angles Θ A and Θ B being interrelated like in the XXZ case. At low temperatures, degenerate bidirectional fluctuations dominate in the transition region between the antiferromagnetic and spin flop phases Illustration: Probability for tilt angels of neighboring spins at k B T/J = 0.558, = 2.44 Conclusion: narrow disordered phase between antiferromagnetic and spin flop phases like in the XXZ model with hidden tetracritical point at T=0 Holtschneider+WS, PRB(2007); EPJ B(2008)

21 Adding single ion anisotropies to XXZ model Add to the classical XXZ antiferromagnet on a square lattice a quadratic single ion anisotropy term (Holtschneider+WS,PRB(2007); EPJ B (2008)): H si =D (Si z)2 i D > 0 introduces a competing planar anisotropy, and stabilizes BC structures, D < 0 enhances the uniaxiality of the exchange anisotropy, and suppresses BC structures. Phase diagram for D/J = 0.2, = 0.8 Biconical phase bordered by antiferromagnetic and spin flop phases with tilt angles depending now on H at fixed T

22 Tilt angle histograms =1.79 =1.81 =1.83 = p(θ) Θ Histograms for probability density of having a spin with the tilt angle Θ at an arbitrary site of the lattice, in the case of a competing single ion term, D/J = 0.2, at k B T/J = 0.2 and varying field H. Note there are two peaks (Θ A, Θ B ) at given H in the BC phase.

23 Phase diagram for D < H c1 spin-flop paramagnetic antiferromagnetic k B T/J Phase diagram for a single ion anisotropy enhancing the uniaxiality of the XXZ model, D/J = 0.2, removing the degeneracy in biconical structures at H c1 and giving rise to a direct transition between AF and SF phases at low temperatures

24 Adding single-ion term on cubic lattice Add to the classical XXZ antiferromagnet on a cubic lattice a single ion anisotropy term (WS,PRE (2013)): H si =D (Si z)2 i D > 0 introduces a competing planar anisotropy, and stabilizes BC structures BC SF AF k B T/J P Phase diagram for D/J = 0.2, = 0.8 Bicritical AF-SF-P and the multicritical AF-BC-SF points, connected by a (dashed) line of first-order transitions

25 S=1 anisotropic Heisenberg chain at T=0 Consider Heisenberg chain with (Tonegawa et al., PTP (2005); Sengupta+Batista, PRL(2007); Peters,McCulloch+WS, PRB(2009,2012) H= (J(Si xsx i+1 +Sy i Sy i+1 + Sz i Sz i+1 )+D(Sz i )2 BSi z) i Uniaxial exchange anisotropy > 1, field B in the standard notation; D quadratic single ion anisotropy, i lattice site. In the ground state, one observes antiferromagnetic (AF), spin-flop (SF) (or, in terminology of quantum magnets: spin-liquid (SL)), biconical (BC) (or supersolid (SS)), and (10) (or IS2), with magnetization plateau at half saturation, phases. Matsubara+Matsuda (1956), Matsuda+Tsuneto (1970), and Liu+M.E. Fisher (1973) noticed: Anisotropic Heisenberg antiferromagnet corresponds to quantum lattice gas, with (i) AF phase corresponding to crystalline phase, (ii) SF phase to superfluid phase, and (iii) BC phase to supersolid phase with non-vanishing AF (crystal) and SF (superfluid) order parameters

26 Ground state phase diagram at D/J = /2 Δ HP SL IS1 SS IS2 FP B J Phase diagram based on quantum MC simulations (SSE) by Sengupta+Batista,PRL(2007); confirmed and extended in DMRG calculations by Peters,McCulloch+WS,PRB(2009) AF SF BC FP Ground state phase diagram of corresponding classical spin chain; no half-magnetization-plateau phase (IS2 or (10)) B/J Supersolid phase is stable in a substantially reduced region compared to the biconical phase.- Distinct types of SL structures:

27 Two distinct types of spin-liquid structures <S z i > M=22 M=25 M= site i Magnetization profiles in the spinliquid phase with total magnetization smaller than half system size for = 2D/J = 3.5, with a (dominant) broad plateau: commensurate case <S z i > M=34 M=48 M= site i Magnetization profiles in the SL phase with total magnetization M larger than half system size L for = 2D/J = 3.5, showing (dominant) oscillatory behavior: incommensurate case, q = 2π(1 m); m = M/L

28 Ground state phase diagram at = 5.0 D/J SL C 10 SL IC SS AF F IC B/J Phase diagram based on DMRG calculations by Peters, McCulloch +WS,JPhys:CS(2010),PRB(2012) D/J SF BC AF F B/J Ground state phase diagram of corresponding classical spin chain Supersolid (SS) phase appreciably reduced compared to BC phase; commensurate and incommensurate spin-liquid (SL)structures; IC phase with exponentially decaying transversal correlations has no classical analogue

29 Conclusions XXZ Heisenberg antiferromagnets in a field along the easy axis classical: square lattice: highly degenerate ground state with biconical structures; separate AF and SF transitions down to very low temperatures on square lattice; hidden tetracritical point at T=0. cubic lattice: evidence for bicritical point of Heisenberg symmetry quantum: S=1/2 XXZ model on square lattice: quantitative corrections to previous analysis; no hint for biconical (BC) structures Adding single ion anisotropies Classical XXZ model plus quadratic single ion anisotropy on square lattice: stable BC phase For the cubic lattice: bicritical point plus multicritical AF-BC-SF point at lower temperature Quantum anisotropic Heisenberg chain plus quadratic anisotropy at T = 0. Comparison to classical case; evidence for commensurate and incommensurate spin-liquid (SF) structures