Neutron Scattering and Magnetism
Laboratory for Solid State Physics · D-PHYS

Defective quantum magnets

Take a "classical" magnetic system, say an ordinary antiferromagnet, and create a defect by removing a single spin. The remaining spins stay aligned in the Néel structure as if nothing happened. Not so in a quantum magnet, where even in the ground state all spins are subject to collective zero-point fluctuations. Removing one spin drastically alters the fluctuations of its remaining neighbors, and the most unusual phenomena may arise as a result.

Best of all, the problem is directly accessible to experiment: replacing a small fraction of the magnetic ions with non-magnetic ones can usually be incorporated into the crystal growth process.

Spin islands in a depleted spin ladder

In-gap spin island scattering in depleted DIMPY Spin island cartoon

Left: neutron spectrum of DIMPY depleted with 4% of non-magnetic zinc. Below the intense magnon band of the host ladder, a weak column of in-gap scattering from the interacting spin islands extends all the way to zero energy [1]. Right: cartoon of a single spin island: a cloud of staggered polarization around the non-magnetic impurity (gray), carrying a net total spin of 1/2.

Take a quantum spin system with a non-magnetic ground state protected by an energy gap, say a spin ladder, and remove a few spins. Paradoxically, what remains becomes magnetic and gapless. Every removed spin leaves its former singlet partner unpaired, and the short-range correlations of the host spread this liberated degree of freedom over an extended region: a "spin island" of staggered polarization, pinned to the impurity but reaching over many unit cells. And since the islands are that large, they overlap: the intervening quantum-disordered medium carries effective interactions between them, exponentially decaying with distance and alternating in sign, quite reminiscent of the RKKY coupling in metals. The impurities thus spawn an entirely new mesoscopic magnetic system, living inside the old one.

The topology of the lattice is what decides how interesting this system gets. In a simple chain every impurity severs the host, and the liberated spins at the fragment ends merely pair up into isolated dimers. A ladder is a different matter: an impurity on one leg leaves the other intact, magnetic continuity is preserved, and a typical fragment hosts many mutually interacting islands. Our material of choice is the strong-leg spin ladder DIMPY, which is easily depleted by substituting the S = 1/2 copper with non-magnetic zinc during crystal growth [1]. Thanks to the long correlation length of this particular ladder, the magnetization profile of a single island extends over some 15 unit cells: an emergent magnetic object roughly 20 nm in size.

The payoff is a parameter-free description of a disordered many-body system. Representing each island by an effective spin-1/2 at a random position, with the numerically calculated sign-alternating couplings, and exactly diagonalizing a mere dozen such spins reproduces the measured susceptibility and magnetization of macroscopic samples in complete quantitative detail [1]. The islands' own collective dynamics is observed directly: high-resolution neutron spectroscopy reveals a weak continuum of excitations inside the spin gap of the host ladder. Note that the momentum linewidth of this scattering is nothing other than the "form factor" of the spin island, an actual image of its real-space shape.

LT scaling

Measured and calculated phase diagrams of depleted spin ladders

Magnetic ordering phase diagrams of the depleted spin ladders BPCB (left) and DIMPY (right) for zinc concentrations z = 0, 0.005 and 0.01. Large circles: experimental transition temperatures from millikelvin calorimetry. Small dots and shaded domes: quantum Monte Carlo predictions. Left inset: measured and calculated susceptibility at 8 T. Right inset: relative suppression of the ordering temperature with impurity concentration, experiment against theory.

A quantum critical system has no energy scale of its own. In a magnetized spin ladder, a realization of the Tomonaga-Luttinger liquid discussed on our quantum criticality page, the only scale is the temperature itself, and every property is a power law in T. Now sprinkle in some defects. Their average spacing L introduces a new length, the spectrum at the critical wave vector is discretized in steps of v/L, and the pristine T-universality gives way to a new "LT universality": the critical susceptibility becomes a universal function of the product LT. This much was known for spin chains, where every impurity chops the liquid into disconnected fragments. But is LT scaling a generic property of any "dirty" quantum liquid, even one whose continuity the defects do not sever, and for arbitrary strength of interactions?

We are, first and foremost, an experimental group. This question, however, we settled by doing the theory in house [1]. Combining analytic calculations with large-scale quantum Monte Carlo simulations, we derived a generalized scaling law for the critical staggered susceptibility of a depleted spin ladder, in which the Luttinger parameter K enters both the exponent and the scaling function itself. The simulations confirm it across the board: for the attractive liquid realized in DIMPY (K = 1.23) as well as the repulsive one in BPCB (K = 0.93), and even though the defects in a ladder remain partially "transparent" to spin correlations. LT universality is thus not an artifact of chopped-up chains; it is generic to dirty Tomonaga-Luttinger liquids.

Of course, a theory is only worth as much as the measurements it predicts. The critical susceptibility is precisely what determines whether the weak residual couplings between ladders can trigger three-dimensional magnetic order, so our scaling law translates directly into a prediction of the entire field-temperature phase diagram of a depleted ladder, and of how fast impurities suppress the ordered phase. We grew series of BPCB and DIMPY single crystals with up to a few percent of zinc and mapped their phase boundaries calorimetrically down to millikelvin temperatures. As the figure shows, the measured and computed phase diagrams agree in complete quantitative detail, for both materials and across the entire field range [1].