# Integer Quantum Magnon Hall Plateau-Plateau Transition in a Spin Ice Model

###### Abstract

Low-energy magnon bands in a two-dimensional spin ice model become integer quantum magnon Hall bands under an out-of-plane field. By calculating the localization length and the two-terminal conductance of magnon transport, we show that the magnon bands with disorders undergo a quantum phase transition from an integer quantum magnon Hall regime to a conventional magnon localized regime. Finite size scaling analysis as well as a critical conductance distribution shows that the quantum critical point belongs to the same universality class as that in the quantum Hall transition. We characterize thermal magnon Hall conductivity in disordered quantum magnon Hall system in terms of robust chiral edge magnon transport.

Bosonic analogue of integer quantum Hall states have been proposed in a number of quasi-particle boson systems with broken time-reversal symmetry such as photon, haldane08 ; raghu08 ; wang09 ; ao09 ; ochiai09 ; lu14 phonon, prodan09 exciton, yuenzhou14 exciton-polariton, karzig14 triplon, romhanyi15 magnon shindou13a ; shindou13b ; shindou14 ; LifaZhang13 ; mook15 ; chisnell15 ; roldan-molina ; kim16 ; owerre and surface magnon-polariton. ochiai16 Typically, their quasi-particle excitations have extended bulk bands with topological integers and topological edge modes whose chiral dispersions cross band gaps among these bulk bands. Due to its chiral (unidirectional) nature, a quasi-particle boson flow along the edge mode is believed to be robust against generic elastic backward scatters, fostering a rich prospect of their future applications. raghu08 ; haldane08 ; wang09 ; ao09 ; ochiai09 ; lu14 ; prodan09 ; yuenzhou14 ; karzig14 ; shindou13a ; shindou13b ; shindou14 ; ochiai16 On the one hand, these bosonic systems often break conservation of the quasi-particle number even at the level of respective quadratic Hamiltonian. yuenzhou14 ; karzig14 ; shindou13a ; shindou13b ; shindou14 ; LifaZhang13 ; mook15 ; chisnell15 ; kim16 ; engelhardt15a ; furukawa15 ; galilo15 ; bardyn16 ; engelhardt15b ; peano16 ; xu16 Thereby, one naturally wonders if the quasi-particle flow along the topological edge modes is still robust against such particle-number-non-conserving perturbations or not. In other words, one may raise a question whether two quantum Hall regimes with different Chern integers are topologically distinguishable even in the absence of the U(1) symmetry associated with the quasi-particle number conservation.

In this rapid communication, we study effects of generic disorder potentials in a simplest spin model in a quantum magnon Hall regime. Our numerical results and the following argument clarify that, even without the explicit U(1) symmetry at the Hamiltonian level, the topological magnon edge mode provides a robust quantized magnon conductance and therefore quantum magnon Hall regimes with different topological integers are always distinguished by a quantum critical point with delocalized bulk magnon band. Thermal conductance distributions calculated at the critical point clearly shows that the quantum critical point belongs to the same universality class as the two-dimensional integer quantum Hall plateau-plateau transition. Based on these knowledge, we give a generic expression for the thermal Hall conductivity in disordered integer quantum bosonic Hall systems from edge transport picture.

We study spin excitations in a square-lattice spin ice model wang06 ; budrikis10 ; budrikis12 ; iacocca16 under out-of-plane Zeeman field ;

(1) |

The model consists of two inequivalent spins in a unit cell, -sublattice spin on the -link of the square lattice and -sublattice spins on the -link. supple Due to a magnetic shape anisotropy, wang06 ; budrikis10 ; budrikis12 ; iacocca16 ; supple each sublattice spin has an easy-axis anisotropy along respective spatial direction. Heisenberg spins are coupled with each other by magnetic dipole-dipole interaction, denotes the unit vector connecting sites and . An inclusion of the next and the next nearest neighbor magnetic dipolar couplings imposes so-called two-in two-out ice rule for each vertex, which has been experimentally observed in a patterned ferromagnetic film. wang06

When the classical ground-state spin configuration becomes fully polarized by the Zeeman field (), the lowest magnon band and the second lowest magnon band acquire the topological number with opposite sign due to the finite next nearest neighbor dipolar coupling, and a topological chiral edge mode appears inside a band gap between the two. supple The corresponding magnon Hamiltonian is obtained from Eq. (1) with , , as,

(2) |

where , denote the nearest and the next nearest dipolar interaction, respectively with and . and are the primitive lattice vectors of the square lattice and (). supple Short-ranged randomness are introduced in Eq. (2); and are uniformly distributed within and . We set the unit of energy to be and that of length the lattice spacing. Due to magnetic anisotropy term, dipolar interaction and randomness, the quadratic boson Hamiltonian does not have any continuous U(1) symmetry associated with magnon number conservation.

Using the transfer matrix method, mackinnon83 ; ohtsuki04 ; kramer05 we first calculated the localization length of a single-particle eigenstate of a corresponding generalized eigenvalue problem with the randomness. Due to the bosonic nature, the eigenvalue problem takes a form of , with and . in the right hand side is a 2 by 2 diagonal Pauli matrix in the particle-hole space; and is an eigenenergy to which belongs. We consider a quasi-one-dimensional (q1d) geometry, where the system is spatially larger in one direction (-direction) than in the other (-direction). For every () with , the system has a finite width along the -direction; with . With and , the generalized eigenvalue equation takes a following matrix form;

(3) |

and . Note that differs from one another for different due to the on-site randomness, while are the same for different with . An by matrix has a symplectic feature; , with being a Pauli matrix in the 2-dimensional space subtended by and . Thus, has a pair of two positive eigenvalues; . The same holds true for . Call a set of all eigenvalues of an Hermitian matrix as with . For sufficiently large , all real converge into finite values (Lyapunov exponents; LE). oseledec68 Using the Gram-Schmidt orthonormalization, we numerically obtained the smallest LE of () for larger ; . is nothing but the largest localization length of the eigenstate of the q1d system at energy . mackinnon83 We set inside the topological band gap in the clean limit.

With weaker randomness, the localization length normalized by decreases on increasing , suggesting that eigenstates in this regime are all localized due to the topological band gap (quantum magnon Hall regime). The same observations hold true with much stronger randomness, indicating that eigenstates in much stronger disordered region belong to a conventional Anderson localized regime. Obtained numerical result (Fig. 1) shows that these two localized regions are always separated by a quantum phase transition point where the normalized localization length barely changes as a function of . The scale invariant behaviour of suggests the existence of a quantum phase transition similar to an integer quantum Hall plateau-plateau transition. huckestein95 ; kramer05

To confirm this, we further calculate the two-terminal magnon conductance for the system from a transmission matrix as . The transmission matrix is calculated from the transfer matrix; , with and . Here we choose to be eigenstates of a model of decoupled one-dimensional chains; . The conductance along the -direction is calculated both with open () and with periodic boundary conditions () along the -direction.

thus calculated tends to have a finite quantized plateau in the quantum magnon Hall regime in the thermodynamic limit (), while showing zero conductance in the conventional localized regime (Fig. 2). The quantization in the quantum magnon Hall regime demonstrates a robust unidirectional magnon transport along the topological chiral edge mode. The bulk conductance seen by tends to have a finite value only at the transition point, while zero otherwise in larger system size. These observations lead to the conclusion that the quantum magnon Hall regime with the robust chiral edge mode and the conventional Anderson localized regime without the edge mode are topologically disconnected by a direct transition point with a delocalized bulk state. Importantly, this holds true irrespectively of the presence of the explicit U(1) symmetry at the Hamiltonian level.

Robustness of chiral magnon edge transports against boson-number-non-conserving elastic perturbations is a consequence of the energy conservation. Our BdG type Hamiltonian has a particle-hole symmetry, , where exchanges particle and hole indices; . Due to this generic symmetry, any eigenstate of has its particle-hole counterpart . A local perturbation which does not conserve the boson number can have a finite matrix element between these two, e.g. with . Physically, however, the hole state and the particle state are different number states of the same quasi-particle excitation, i.e. and , and the scattering process between these two is accompanied by an energy emission (or absorption) of , where is an energy quantum for the quasi-particle excitation; and . Thus, any magnon state with cannot be scattered into its hole counterpart by elastic scattering. In other words, particle and hole channels are completely decoupled both in the transmission matrix and in a reflection matrix in the two-terminal conductance calculation above. This results in the robustness of the chiral magnon edge transport even in the presence of boson-number-non-conserving perturbations. The decoupled nature of particle and hole channels also allows to define a magnon current even in the absence of the explicit U(1) symmetry in the magnon Hamiltonian; the magnon continuity equation without the source term can be derived from the equation of motion for the Green function as far as elastic scattering is concerned.

The robust chiral edge conductance in the quantum magnon Hall regime indicates that the Hall regime with the quantized edge conductance () is always disconnected from the conventional localized regime by a direct transition with delocalized bulk states. halperin82 This is indeed the case with a phase diagram subtended by the disorder strength and the single-magnon energy (Fig. 3). For a fixed , the Hall regime is encompassed by the two direction transition points at and . For , is quantized and vanishes in the thermodynamic limit. For the other region, both and tend to vanish in a larger system size. supple

A finite-size scaling analyses of (of Fig. 2) near the transition point () is carried out based on , with the critical exponent, a scaling dimension of a leading-order irrelevant scaling field at the critical point and fitting parameters. slevin14 For , the 95% confidence interval of is [2.28,2.60] with goodness of fit . By omitting the smallest size, the estimate is [2.54, 2.86] with goodness of fit . Though being consistent with the recent estimate of of the quantum Hall university class (), slevin09 ; obuse10 ; amado11 ; fulga11 ; dahlhaus11 ; obuse12 the error bars are too large to conclude this affirmatively. To this end, we further calculated distributions of the conductances at the critical point (Fig. 4). The distributions have striking similarities to the critical conductance distributions of the two-dimensional Chalker-Coddington network model, chalker88 which strongly suggests that the direct transition belongs to the quantum Hall universality class.

Based on these knowledge, let us finally characterize an edge-mode contribution to thermal magnon Hall conductivity in the disordered quantum magnon Hall regime. To this end, we impose an open/periodic boundary condition along the /-direction, introduce a temperature gradient along the -direction, and calculate an energy current along the -direction. The energy Hall current is given as a function of the disorder strength . For a given , the system has sub-extensive number of chiral edge modes within , where . Thus, the edge modes within give a magnon Hall current density of , with the system size along the -direction. The energy Hall current density due to these chiral edge modes around with higher temperature and that around with lower temperature are therefore,

respectively with Bose function . A sum of these two is proportional to the temperature gradient ; . The thermal Hall conductivity takes a form;

(4) |

with and is a non-analytic function; . The above argument can be easily generalized into generic quantum magnon Hall systems with disorders. supple Note also that the edge-mode contribution dominates total in a system with the quasi-one-dimensional geometry (), where a bulk contribution diminishes as ( being a constant of the order of ) due to the localization effect in one-dimensional systems.

In this rapid communication, we studied low-energy magnon bands in a two-dimensional spin ice model with disorders. We show that the magnon bands with disorders undergo a direct transition from an integer quantum magnon Hall regime to a conventional magnon localized regime. The critical conductance distributions at the transition point suggest that the direct transition belongs to quantum Hall universality class. The obtained result can be tested by standard microwave antennas experiments. supple Based on the edge magnon transport picture, we give a generic expression for thermal magnon Hall conductivity in disordered quantum magnon Hall systems. The obtained expression is qualitatively consistent with an expression of thermal magnon Hall conductivity in the clean limit, previously obtained based on the linear response theory. matsumoto11a ; matsumoto11b ; matsumoto14 ; qin11 ; qin12 ; supple

The authors thank Junren Shi for fruitful discussions. This work was supported by JSPS KAKENHI Grants No. 15H03700 and No. 24000013 and by NBRP of China (Grant No. 2015CB921104).

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## I Supplemental Materials for “Integer Quantum Magnon Hall Plateau-Plateau Transition in a Spin Ice Model”

## Ii two-dimensional spin ice model under strong out-of-plane field

A magnetic system considered consists of two inequivalent ferromagnetic ‘islands’ of the order of 100 nm size s-wang06 ; one is centered on a -link of a two-dimensional square lattice and the other is on a -link (Fig. 5). The ferromagnetic island on the /-link is spatially elongated along the /-direction respectively. Thus, their magnetic moments prefer to point along the -direction respectively due to the magnetic shape anisotropy. We model these two ferromagnetic islands as two inequivalent spins on the -link with large magnetic moment (/ respectively). The shape anisotropy is included as an effective single-ion spin-anisotropy energy such as and . Ferromagnetic moments are coupled with one another via the magnetic dipole-dipole interaction s-wang06 , so do spins in the spin model. Based on the Holstein-Primakoff mapping, the corresponding quadratic magnon Hamiltonian is derived as in Eq. (2) (in the main text). For simplicity, we include only dipole couplings between the nearest neighbor spins (denoted by : see Fig. 5) and between next nearest neighbor spins (denoted by or : see Fig. 5). The ratio among these three are chosen to be consistent with the dipolar coupling strength (see below).

Without randomness (), the quadratic Hamiltonian given in Eq. (2) in the main text can be Fourier-transformed,

(5) |

with

(6) |

where and . 2 by 2 Pauli matrix is for the particle-hole space, while 2 by 2 Pauli matrix is for the A and B sublattices. Coefficients in Eq. (5) are

(7) | ||||

(8) | ||||

(9) |

where we put for simplicity. The 4 by 4 matrix is diagonalized at every by a paraunitary transformation s-colpa78

(10) |

where , , , and are defined by , , , and as follows;

(11) | ||||

(12) | ||||

(13) | ||||

(14) | ||||

(15) |

with

(16) |

In terms of the transformation, the BdG Hamiltonian is paraunitary equivalent to a diagonal matrix,

where the two magnon energy bands are even functions in ;

(17) |

Note that is sufficiently large that the lower magnon band is fully gapped; for . This also allows that and . In the absence of the next-nearest neighbor dipolar interaction (), two bands form a band touching with a massless Dirac dispersion at and , where . The Dirac dispersions acquire a finite mass by non-zero next-nearest neighbor dipolar interaction and the sign of the mass is determined by that of . Due to this mass acquaintance, the upper magnon band () and lower magnon band () have Chern integer respectively for . When changes its sign from positive to negative, these two integers change into respectively.

To calculate the Chern integer for the upper magnon band directly s-thouless82 ; s-kohmoto85 ; s-shindou13a ; s-engelhardt15 ; s-furukawa15 ; s-xu16 , look into the first column of the paraunitary matrix , which is nothing but a periodic part of the Bloch wavefunction for the upper magnon band,

(18) |

where and ( and ) are connected with each other by the rotation (, exchanges A and B sublattices). and are connected with each other by a particle-hole transformation, which is generic in any quadratic boson Hamiltonian; . is given by

(19) |

while the other three are obtained from this by the rotation or by the particle-hole transformation. is given by a proper normalization; . By using Eqs .(7-16), one can see that (and also ) has a zero only at , while and have a zero at . Thus, we expand with respect to small with ;

(20) |

where , . and are calculated as follows,

for and . Note that . Thus, a phase of , i.e. with , acquires phase holonomy, whenever rotates once around anti-clockwise. This dictates that the Chern integer for the upper band is (that for the lower band is due to the sum rule s-shindou13a ). The non-zero topological integers for these two magnon bands result in a topological chiral magnon edge mode within the band gap s-halperin82 ; s-hatsugai93 ; s-shindou13a ; s-engelhardt15 ; s-furukawa15 ; s-xu16 . The sign of the integer dictates that the mode has a chiral dispersion with the anti-clockwise rotation when viewed from direction and the out-of-field is along direction (“right-handed” chiral mode). So far, we assume that . For , we confirmed numerically that the same band gap with the chiral edge mode persists (Fig. 6).

## Iii A phase boundary between the quantum magnon Hall regime and conventional magnon localized regime

A phase diagram in the main text (Fig. 3) has the quantum magnon Hall regime and conventional magnon localized regime. The boundary between these two regions is identified as a scale-invariant point of the two-terminal conductance calculated with the periodic boundary condition; . Fig. 7 shows as a function of the single-particle (magnon) energy for several . Thereby, we found two such scale-invariant points (one at specified by a black dotted line in Fig. 7 and the other at by a red dotted line). For an “edge conductance” characterized by has a tendency to take the quantized value () in the thermodynamic limit ( is the conductance along the -direction with the open boundary condition along the -direction) . For or , the edge conductance goes to zero. From these observations, we regard the former region as the quantum magnon Hall regime and the latter as the conventional magnon localized regime.

## Iv microwave antennas experiment

The two terminal magnon conductance calculated in the main text can be measured in a standard microwave experiment commonly used for spin wave experiments s-serga10 . The experiment consists of two microstrip microwave antennas attached to the two-dimensional square-lattice spin ice system (Fig. 8). The two antennas are spatially separated from each other shorter than a spin coherent length, over which spin wave propagates without an energy dissipation. Note that the spin coherent length in ferromagnetic insulator such as YIG can be over millimeters, while it is at most on the order of several micrometer in ferromagnetic metals.

The role of the first antenna is for spin wave excitation and that of the second antenna is for its detection. An a.c. electric current with a frequency in the microwave regime (let us call this as ‘external frequency’) is introduced in the first antenna (‘input signal’). The current locally excites spin wave with the same external frequency. The spin wave propagates through the magnonic crystal system, and, after a certain time, the spin wave reaches the second antenna, where an a.c. electric current is induced (‘output signal’).

The two terminal magnon conductance studied in the main text corresponds to a transmission ratio between the output electric current and input current. The ratio can be obtained as function of the external frequency within the microwave regime s-serga10 . When the frequency and the disorder strength are chosen inside the quantum magnon Hall regime, the transmission ratio is finite (see Fig. 3 of the main text; the frequency corresponds to energy of magnon states in the figure). Especially, the ratio is dominated by chiral edge spin wave transport, when the distance between two antennas is longer than the localization length. Namely, the bulk spin wave excited by the first antenna dies off quickly before it reaches the second antenna due to its finite localization length. Meanwhile, the chiral edge spin wave excited by the first antenna travels along the edge without being backward scattered.

When the frequency and the disorder are in the conventional magnon localized regime, the transmission ratio reduces dramatically. The ratio becomes exponentially small, if the localization length is much shorter than the distance between the two antennas. Accordingly, the quantum phase transition from the quantum magnon Hall regime to conventional magnon localized regime can be experimentally measurable through the dramatic reduction of the transmission ratio as a function of either the external frequency or the disorder strength.

Note that the distance between the two antennas must be shorter than a finite spin coherence length (Fig. 8). The finite distance between the two antennas may result in a blurred change of the transmission ratio at the phase transition point. Nonetheless, the spin ice model made out of ferromagnetic insulator such as YIG allows a very large distance between the two antennas, e.g. 8mm in YIG s-serga10 . Since a typical localization length would be at largest on the order of micrometer scale s-evers15 , the very large spin coherence length in YIG may even enable us to study the critical properties of the quantum phase transition.

## V thermal magnon Hall conductivity in generic disordered quantum magnon Hall systems and its relation to the thermal magnon Hall conductivity in the clean limit

In the main text, we have studied only the model with two magnon bands. A realistic material may have more than two magnon bands, which have non-zero quantized Chern integers. Our study as well as established knowledge on interplays between localization effect and quantum Hall physics s-prange suggests that even small disorder makes all these bulk magnon bands localized except for delocalized bulk states at respective band center (Fig. 9(a,b)). A pair of two delocalized bulk states bound a mobility gap, inside which a topological chiral edge mode lives (Fig. 9(b)). For the two-band model studied in the main text, the bulk delocalized states at and encompass the mobility gap, inside which the chiral edge mode lives. As in Fig. 3 of the main text, the edge mode disappears when a pair of the two delocalized bulk states fall into the same energy.

For the generic situation described above, we can employ the same argument as in the main text, to derive an edge contribution to the thermal Hall conductivity,

(21) |

Here the summation is taken over chiral edge modes; the integer counts chiral edge modes at finite frequency. and stand for a pair of two energies by which the -th chiral edge mode is bounded (Fig. 9(b)). We define when the -th chiral edge mode is right-handed, while when the mode is left-handed. and is the Bose distribution function.

The above expression is qualitatively consistent with the thermal magnon Hall conductivity in the clean limit, which was previously obtained based on the linear response theory s-matsumoto11a ; s-matsumoto11b ; s-matsumoto14 ; s-qin11 ; s-qin12 ;

(22) |

Here