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

Quantum-critical dynamics

Is it possible to make quantitative predictions for experiments based on only qualitative information about the material under study? Normally, of course, not. Yet under so-called critical conditions, often realized near phase transitions, the impossible becomes routine: systems with entirely different microscopics but a few qualitative similarities show quantitatively identical (universal) behavior at long wavelengths and long time scales. The details simply cease to matter. Quantum criticality, which occurs in the limit of very low temperatures, is the most spectacular case of all: there a single universality class dictates not only the static properties but the entire dynamics.

Case in point: one dimensional spin systems. Their low-temperature low-energy long-wavelength properties are universal regardless of the details of magnetic interactions. Even better, they are identical to those of interacting impenetrable Bosons, which in one dimension can famously be mapped to Fermions. For the Fermionic model there are exact analytical predictions for the specific heat, density and correlation functions, with the interaction potential dropping out entirely. These results carry over verbatim to spin systems, where they describe the measured heat capacity, magnetization and inelastic neutron scattering. A theory with no adjustable parameters, confronted with experiments on real materials: we have performed a whole series of such tests in quasi-one-dimensional quantum magnets [1].

Scale factor-free universality

Excitation spectrum of BPCB near the critical field Specific heat scaling in BPCB

Left: magnetic excitation spectrum of the spin ladder compound BPCB near the critical field of magnon condensation. The two intense dispersive bands at high energies are Zeeman-split triplet excitations with their bound states. The yellow arrow marks the non-universal longitudinal excitations; the red dashed box encloses the universal critical fluctuations of interest. Right: scaling plot of the magnetic specific heat. Data collected in fields between 4 and 9 T (symbol colors) collapse onto the exact free-Fermion scaling function (solid line), drawn with no adjustable parameters whatsoever; a = 1/2 and zν = 1 are the universal critical exponents. Inset: quality of the data collapse when the two exponents are allowed to vary.

At a generic phase transition, universality delivers the critical exponents, but the scaling functions themselves are usually beyond the reach of exact theory. The field-induced Bose-Einstein condensation of magnons in one dimension is a rather glorious exception. At this quantum critical point the density of magnons vanishes, and in one dimension a dilute gas of hard-core Bosons is indistinguishable from free Fermions. Everything, thermodynamics and correlation functions alike, can then be computed exactly, with no adjustable parameters, not even an overall scale factor. This is an almost unique situation in many-body physics: a strongly correlated system whose finite-temperature dynamics is known in complete quantitative detail. We were, in fact, the first to numerically evaluate the dynamic scaling function [1]. All that remained was to measure it.

Easier said than done. Our first attempt, on the spin chain material K2CuSO4Cl2, largely failed [1]. The universal fluctuations are those transverse to the applied field, and near the transition they are swamped by strong non-universal longitudinal scattering at exactly the same energies. Unpolarized neutrons cannot tell the two apart, and a polarized experiment on this scale is infeasible to this day. The solution was a "trick": replace the chain by a spin ladder. A ladder possesses a symmetry that a chain simply does not have, the exchange of its two legs. The critical fluctuations are odd under this exchange, the parasitic ones are even, and the two channels modulate the neutron intensity differently across reciprocal space. The material itself thus acts as a built-in "symmetry filter", and no polarized neutrons are needed.

In the strong-rung ladder compound BPCB this program succeeded completely [2]. The measured magnetic specific heat collapses onto a single curve that is the exact free-Fermion scaling function, with nothing fitted at all. The local dynamic structure factor, extracted from the leg-odd channel, follows the exact prediction over one and a half decades in ħω/kBT. The finite-temperature correlation functions of one-dimensional hard-core Bosons were thereby measured for the first time, in any system.

Tomonaga-Luttinger spin liquids

Enlarged view: Scaling of critical temporal spin correlations in the magnetized phase of DIMPY. The solid line is the exact analytiycal result for attractive Fermions result expressed through Euler's Gamma-functions below [3].

Scaling of critical temporal spin correlations in the magnetized phase of the spin ladder DIMPY, measured by inelastic neutron scattering at temperatures from 59 mK to 5.3 K [2]. The solid line is the exact analytical result for attractive Fermions, given by the formula below in terms of Euler's Gamma functions. Inset: quality of the data collapse versus the scaling exponent γ.

In three dimensions, interacting electrons are handled by Landau's Fermi liquid theory: the excitations are electron-like quasiparticles, only slightly "dressed" by the interactions. In one dimension this picture collapses entirely. A particle cannot move without shoving all of its neighbors, so no individual quasiparticles survive; every excitation is collective, a density wave running along the line. What replaces the Fermi liquid is the Tomonaga-Luttinger liquid, the universal low-energy state of essentially any gapless one-dimensional system.

The truly remarkable part is the economy of the description. All low-energy properties of a Tomonaga-Luttinger liquid are controlled by just two numbers: the velocity u of the collective waves, and the dimensionless Luttinger parameter K that encodes the interactions (K = 1 for free Fermions, K > 1 for attraction, K < 1 for repulsion). It does not matter in the least what the system is actually made of: a one-dimensional metal, cold atoms in an optical trap, or a quantum spin chain, where the role of the particles is played by flipped spins. Once u and K are fixed, universality does the rest: the thermodynamics, the correlation and response functions all follow, often in closed analytical form. The rather formidable-looking formula in the figure is a case in point: it is the exact finite-temperature dynamic structure factor of a magnetized spin chain or ladder, expressed through Euler's Gamma functions, with K as the only "knob" in sight.

Quantum magnets let us put this economy to the test. A partially magnetized Heisenberg spin chain, such as realized in 2(1,4-Dioxane)·2(H2O)·CuCl2, is a Tomonaga-Luttinger spin liquid equivalent to repulsive Fermions [1]. A partially magnetized strong-leg spin ladder, realized in the compound DIMPY, gives Fermions with attraction [2]. For the latter we measured the critical dynamics by inelastic neutron scattering [2] and NMR [3], and the static critical properties by calorimetry and magnetometry [3]. As the figure shows, spectra spanning two decades in temperature collapse onto the predicted universal curve, with the best-fit scaling exponent of −0.60 against the theoretical value of −0.58. Note that beyond the two numbers, the theory receives no input about the actual spin Hamiltonian whatsoever. The agreement is staggering.