Landau theory of multipolar orders in $\mathrm{Pr}{(Y)}_{2}{X}_{20}$ Kondo materials ($Y=\mathrm{Ti}$, V, Rh, Ir; $X=\mathrm{Al}$, Zn)

Landau theory of multipolar orders in $\Pr {(Y)}_{2} X_{20}$ Kondo materials ( $Y = Ti$ , V, Rh, Ir; $X = Al$ , Zn)

SungBin Lee, Simon Trebst, Yong Baek Kim, and Arun Paramekanti

Phys. Rev. B 98, 134447 – Published 29 October 2018

Abstract

A series of $\Pr {(TM)}_{2} X_{20}$ (with TM = Ti, V, Rh, Ir and X = Al, Zn) Kondo materials, containing non-Kramers $\Pr^{3 +} 4 f^{2}$ moments on a diamond lattice, have been shown to exhibit intertwined orders such as quadrupolar order and superconductivity. Motivated by these experiments, we propose and study a Landau theory of multipolar order to capture the phase diagram and its field dependence. In zero magnetic field, we show that different quadrupolar states, or the coexistence of quadrupolar and octupolar orderings, may lead to ground states with multiple broken symmetries. Upon heating, such states may undergo two-step thermal transitions into the symmetric paramagnetic phase, with partial restoration of broken symmetries in the intervening phase. For nonzero magnetic field, we show the evolution of these thermal phase transitions strongly depends on the field direction, due to clock anisotropy terms in the free energy. Our findings shed substantial light on experimental results in the $\Pr {(TM)}_{2} {Al}_{20}$ materials. We propose further experimental tests to distinguish purely quadrupolar orders from coexisting quadrupolar-octupolar orders.

Received 16 June 2018
Revised 9 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.134447

Physics Subject Headings (PhySH)

Magnetic coupling Magnetic interactions Magnetic phase transitions Magnetism

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

SungBin Lee^1,*, Simon Trebst², Yong Baek Kim^3,4, and Arun Paramekanti^3,4,†

¹Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea
²Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany
³Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
⁴Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada

^*sungbin@kaist.ac.kr
^†arunp@physics.utoronto.ca

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Vol. 98, Iss. 13 — 1 October 2018

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Figure 1
(a) Phase diagram of the Landau theory described by $F_{ϕ u} + F_{ϕ s} + F_{int}^{(3)}$ with $AF Q$ and $F Q$ order parameters as functions of $r_{s ϕ}$ and $r_{u ϕ}$ . Solid (dashed) lines indicate the second (first) order phase transitions. Here, we take $w v λ < 0$ . The various phases are paramagnet (PM), $F Q$ and three distinct phases (I), (II), and (III) for $AF Q$ with parasitic $F Q$ ( $AF Q_{F Q}$ ). See main text for details. (b) Plot of the order parameters as a function of temperature $T$ , a cut through the trajectory (purple line) in panel (a). Red and blue lines represent the magnitude of order parameters $ϕ_{s}$ and $ϕ_{u}$ . (c) Common origin plots of distinct $AF Q_{F Q}$ phases (I), (II), and (III). Red and Blue arrows exhibit $AF Q$ and $F Q$ , respectively, and purple arrow is the direction of quadrupolar order resulting from combination of both $AF Q$ and $F Q$ . All these spin configurations have threefold degeneracies with $2 π / 3$ rotation in $τ_{x} - τ_{y}$ plane.
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Figure 2
(a) Phase diagram of the Landau theory described by $F_{ϕ u} + F_{ϕ s} + F_{m u} + F_{int}^{(3)} + F_{int}^{(4)}$ with $AF Q$ , $F Q$ and $F O$ order parameters as functions of $r_{s ϕ}$ and $r_{s m}$ . Here, we set $w v λ > 0$ distinct with the case depicted in Fig. 1. Thus, the second phase transition only arises from developing additional octupolar order. In this case, three phases exist described by continuous phase transitions: paramagnet (PM), $AF Q$ with parasitic $F Q$ ( $AF Q_{F Q}$ ), and coexisting $AF Q$ and $F O$ with parasitic $F Q$ order ( $AF Q_{F Q}$ $F O$ ). See main text for details. (b) Plot of the order parameters along shown trajectory (purple line) in panel (a). Red, blue and green lines represent the magnitude of order parameters $ϕ_{s}$ and $ϕ_{u}$ and $m_{u}$ . (c) Common origin plots of each phase. Red, blue, and green arrows exhibit magnitudes of $AF Q$ , $F Q$ , and $F O$ phases, respectively, and purple arrow is the combination of them, determining the direction of pseudospin $τ$ . All these spin configurations have threefold degeneracies with $2 π / 3$ rotation in $τ_{x} - τ_{y}$ plane. (Here we chose the quadrupole order configuration having only $τ_{y}$ component.)
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Figure 3
(a) Phase diagram of quadrupolar order as functions of magnetic field $B / / (001)$ and temperature T taking the shown trajectory along the purple line in Fig. 1. In the presence of field $B / / (001)$ , the type (I) phase of $AF Q_{F Q}$ is stabilized at intermediate temperature, whereas the type (III) phase is no longer stable with fields along (001) direction. See the main text for details. (b) Phase diagram of quadrupolar order with $B / / (110)$ . With fields, the type (III) phase is stable favored by both cubic anisotropy and field coupling of $F Q$ .
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Figure 4
(a) Phase diagram of quadrupole-octupole order as functions of magnetic field $B / / (001)$ and temperature T taking the shown trajectory along the purple line in Fig. 2. With fields along (001) direction, the phase transition temperature between $AF Q_{F Q}$ and $AF Q_{F Q} - F O$ decreases, whereas increases for between $AF Q_{F Q}$ and paramagnet. (b) Phase diagram of coexisting quadrupole-octupolar order as functions of magnetic field $B / / (110)$ . In this case, magnetic fields induce the increase of phase transition temperature for both cases. See the main text for details.
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Landau theory of multipolar orders in $\Pr {(Y)}_{2} X_{20}$ Kondo materials ( $Y = Ti$ , V, Rh, Ir; $X = Al$ , Zn)

SungBin Lee, Simon Trebst, Yong Baek Kim, and Arun Paramekanti

Phys. Rev. B 98, 134447 – Published 29 October 2018

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Landau theory of multipolar orders in Pr(Y)2X20 Kondo materials (Y=Ti, V, Rh, Ir; X=Al, Zn)

SungBin Lee, Simon Trebst, Yong Baek Kim, and Arun Paramekanti

Phys. Rev. B 98, 134447 – Published 29 October 2018

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Landau theory of multipolar orders in $\Pr {(Y)}_{2} X_{20}$ Kondo materials ( $Y = Ti$ , V, Rh, Ir; $X = Al$ , Zn)