Maximally Frustrated Magnetic Systems
Fully Frustrated Ladder
The model of a spin ladder whose rungs have equal leg coupling and cross coupling has long been known to have a number of special properties: the spin is a conserved quantity on every rung, there is a first-order quantum phase transition (QPT) between pure rung-singlet and rung-triplet (Haldane) phases, all n-triplet bound states within the singlet phase are exact and they are exactly localised (all excitation bands are completely flat in reciprocal space). We decided that this made it an excellent model on which to begin an investigation of the unconventional properties of quantum magnets with exact (or perfect, or ideal) frustration.
This criterion in fact covers a significant number of models, and it has to be said that we have discovered a significant number of new physical phenomena. One of the keys to our success was the discovery that, when quantum Monte Carlo (QMC) is performed not in the spin basis but in the rung basis, the fully frustrated ladder has no sign problem at all, and thus we were able to compute its thermodynamic properties to extremely high accuracy [1]. In fact all the lowest-lying n-triplet bound states have an energy similar to the one-triplet gap, and this collapse of very many states to the same energy at the QPT leads to a very anomalous evolution of the thermodynamics there. In particular, the energy scale normally taken to characterise the magnetic specific heat and susceptibility becomes far smaller than one expects from the properties of conventional gapped quantum magnets.
Fully Frustrated Ladder Dynamics
Because the ladder always has a robust gap, exact diagonalisation gives a meaningful account of its dynamics at all temperatures. We computed the spectral functions at all temperatures for a number of different coupling ratios, and for ladders ranging from fully frustrated to unfrustrated [2]. We were particularly interested in characterising the shift of spectral weight out of the one-triplet band and into all the multi-triplet excitations, and this showed the same tendency as in the thermodynamic quantities to produce strong effects at anomalously low temperatures. We also found the initially counterintuituve result that the spectrum still has significant structure at infinite temperature, which is in fact a simple consequence of matrix-element effects.
Because the ladder always has a robust gap, exact diagonalisation gives a meaningful account of its dynamics at all temperatures. We computed the spectral functions at all temperatures for a number of different coupling ratios, and for ladders ranging from fully frustrated to unfrustrated [2]. We were particularly interested in characterising the shift of spectral weight out of the one-triplet band and into all the multi-triplet excitations, and this showed the same tendency as in the thermodynamic quantities to produce strong effects at anomalously low temperatures. We also found the initially counterintuituve result that the spectrum still has significant structure at infinite temperature, which is in fact a simple consequence of matrix-element effects.
Sign-Free Quantum Monte Carlo Simulations
Because QMC provides one of the more powerful and versatile numerical approaches to study condensed matter systems at all temperatures, the "sign problem" that arises for frustrated quantum spin models constitutes a serious handicap. The discovery that the sign problem is basis-dependent is therefore a potential game-changer. We generalised our dimer-basis QMC calculations to all partially frustrated two-leg spin-1/2 ladders, meaning those where the diagonal and leg couplings can take any antiferromagnetic values [3]. We found that the sign problem does reappear, but remains remarkably mild throughout the entire phase diagram, and we explained this result in terms of the geometry of operator strings actually containing a minus sign. We applied this result to perform efficient quantum Monte Carlo simulations of frustrated ladders of all coupling ratios, obtaining accurate results for the magnetic specific heat and susceptibility of ladders up to L = 200 rungs and down to very low temperatures.
Because QMC provides one of the more powerful and versatile numerical approaches to study condensed matter systems at all temperatures, the "sign problem" that arises for frustrated quantum spin models constitutes a serious handicap. The discovery that the sign problem is basis-dependent is therefore a potential game-changer. We generalised our dimer-basis QMC calculations to all partially frustrated two-leg spin-1/2 ladders, meaning those where the diagonal and leg couplings can take any antiferromagnetic values [3]. We found that the sign problem does reappear, but remains remarkably mild throughout the entire phase diagram, and we explained this result in terms of the geometry of operator strings actually containing a minus sign. We applied this result to perform efficient quantum Monte Carlo simulations of frustrated ladders of all coupling ratios, obtaining accurate results for the magnetic specific heat and susceptibility of ladders up to L = 200 rungs and down to very low temperatures.
Fully Frustrated Bilayer
We built on these results by extending them to 2D: once again, the fully frustrated S = 1/2
Heisenberg bilayer allows sign-free QMC. Once again, there is a first-order transition from rung-singlet to rung-triplet phases, but now the latter has magnetic order at zero temperature. We used high-precision QMC simulations to establish the finite-temperature phase diagram and demonstrate the presence of a line of first-order transitions terminating at a critical point, which is in the Ising universality class [4]. Further, if the frustration is relaxed back to the well-known Heisenberg bilayer, the continuous transition occurring at zero temperature between the dimer-singlet state and the antferromagnetically ordered bilayer state terminates on the first-order line, giving a quantum critical end point. This regime is not sign-problem-free, and so here we used T = 0 tensor-network calculations to follow the first-order discontinuities.
We built on these results by extending them to 2D: once again, the fully frustrated S = 1/2
Heisenberg bilayer allows sign-free QMC. Once again, there is a first-order transition from rung-singlet to rung-triplet phases, but now the latter has magnetic order at zero temperature. We used high-precision QMC simulations to establish the finite-temperature phase diagram and demonstrate the presence of a line of first-order transitions terminating at a critical point, which is in the Ising universality class [4]. Further, if the frustration is relaxed back to the well-known Heisenberg bilayer, the continuous transition occurring at zero temperature between the dimer-singlet state and the antferromagnetically ordered bilayer state terminates on the first-order line, giving a quantum critical end point. This regime is not sign-problem-free, and so here we used T = 0 tensor-network calculations to follow the first-order discontinuities.
SrCu2(BO3)2 and the Shastry-Sutherland Model
Significant motivation for all of these studies was provided by the Shastry-Sutherland model, which is a S = 1/2 Heisenberg model on an orthogonal dimer geometry formulated specifically because it has an exact dimer-singlet ground state. It also has a near-exact materials realisation in the form of the compound SrCu2(BO3)2. The model has a first-order transition from the gapped dimer-singlet state to a gapped plaquette state, occurring at J'/J = 0.675, and the interaction ratio in the material has been estimated as 0.63 at ambient pressure. The dimer-singlet state was already known to have strongly bound two-triplon states, a high degeneracy of these states and anomalous thermodynamic properties. To connect these properties with the perfect-frustration phenomena of our model studies, we started by linking the bilayer and Shastry-Sutherland phase diagrams and pushing our QMC simulations to their limit, which we found to be J'/J = 0.526 [5].
Significant motivation for all of these studies was provided by the Shastry-Sutherland model, which is a S = 1/2 Heisenberg model on an orthogonal dimer geometry formulated specifically because it has an exact dimer-singlet ground state. It also has a near-exact materials realisation in the form of the compound SrCu2(BO3)2. The model has a first-order transition from the gapped dimer-singlet state to a gapped plaquette state, occurring at J'/J = 0.675, and the interaction ratio in the material has been estimated as 0.63 at ambient pressure. The dimer-singlet state was already known to have strongly bound two-triplon states, a high degeneracy of these states and anomalous thermodynamic properties. To connect these properties with the perfect-frustration phenomena of our model studies, we started by linking the bilayer and Shastry-Sutherland phase diagrams and pushing our QMC simulations to their limit, which we found to be J'/J = 0.526 [5].
To get closer to the QPT it was time for a breakthrough in numerical methods, so we introduced two qualitative advances for the computation of thermodynamic properties. One is the use of thermal pure quantum (TPQ) states, which allows the size of clusters amenable to exact diagonalisation to be augmented dramatically. The second is the use of tensor-network methods, in the form of infinite projected entangled pair states (iPEPS), for representing thermal states. Staying in the dimer-singlet regime of the Shastry-Sutherland model, we demonstrated convergence as a function of system size in TPQ calculations and of bond dimension in our iPEPS results, with complete mutual agreement even extremely close to the QPT [6]. On the physics side, our methods revealed a remarkably sharp and low-lying feature in the magnetic specific heat around the QPT, whose origin appears to lie in a proliferation of excitations composed of two-triplon bound states. The surprisingly low energy scale and apparently extended spatial nature of these states explain why less sophisticated numerical approaches have failed to capture their physics. We obtained the first satisfactory quantitative description of the 20-year-old ambient-pressure specific-heat and susceptibility data, at J'/J = 0.63(1), and the first numerical results capable of describing coupling ratios all the way to the QPT.
Why one might want to perform calculations at the QPT becomes apparent when one applies pressure to SrCu2(BO3)2. Not only does one cross the QPT into the plaquette phase at about 19 kbar, but one's specific-heat measurements show the characteristic broad humps of the gapped dimer and plaquette phases evolve into a very sharp peak at approximately 3 K: the critical point [7]. Finite-temperature iPEPS calculations performed at and on both sides of the QPT confirm the critical point, the sharp spike in C(T), a diverging correlation length at Tc, a discontinuity in the dimer <S.S> at T < Tc that terminates there and the Ising nature of the critical physics. In both experiment and theory, we also added a magnetic field in order to confirm the full picture of the critical-point phase diagram. This led us to discuss the nature of the plaquette phase, which sets in below 2 K, and to find the antiferromagnetically ordered (AFM) phase of the Shastry-Sutherland model at 26 kbar for fields above 3 T.
[1] Thermodynamic Properties of Highly Frustrated Quantum Spin Ladders: Influence of
Many-Particle Bound States,
A. Honecker, S. Wessel, R. Kerkdyk, T. Pruschke, F. Mila and B. Normand,
Phys. Rev. B 93, 054408 (2016).
[2] Multi-Triplet Bound States and Finite-Temperature Dynamics in Highly Frustrated Quantum
Spin Ladders,
A. Honecker, F. Mila and B. Normand,
Phys. Rev. B 94, 094402 (2016).
[3] Efficient Quantum Monte-Carlo Simulations of Highly Frustrated Magnets: the Frustrated
Spin-1/2 Ladder,
S. Wessel, B. Normand, F. Mila and A. Honecker,
SciPost Phys. 3, 005 (2017).
[4] Thermal Critical Points and Quantum Critical End Point in the Frustrated Bilayer Heisenberg
Antiferromagnet,
J. Stapmanns, P. Corboz, F. Mila, A. Honecker, B. Normand and S. Wessel,
Phys. Rev. Lett. 121, 127201 (2018).
[5] Thermodynamic Properties of the Shastry-Sutherland Model from Quantum Monte Carlo Simulations,
S. Wessel, I. Niesen, J. Stapmanns, B. Normand, F. Mila, P. Corboz and A. Honecker,
Phys. Rev. B 98, 174432 (2018).
[6] Thermodynamic Properties of the Shastry-Sutherland Model throughout the Dimer-Product Phase,
A. Wietek, P. Corboz, S. Wessel, B. Normand, F. Mila and A. Honecker,
Phys. Rev. Res. 1, 033038 (2019).
[7] A Quantum Magnetic Analogue to the Critical Point of Water,
J. Larrea Jiménez, S. P. G. Crone, E. Fogh, M. Zayed, R. Lortz, E. Pomjakushina, K. Conder,
A. M. Läuchli, L. Weber, S. Wessel, A. Honecker, B. Normand, Ch. Rüegg, P. Corboz, H. M.
Rønnow and F. Mila,
unpublished (arXiv:2009.14492).
Many-Particle Bound States,
A. Honecker, S. Wessel, R. Kerkdyk, T. Pruschke, F. Mila and B. Normand,
Phys. Rev. B 93, 054408 (2016).
[2] Multi-Triplet Bound States and Finite-Temperature Dynamics in Highly Frustrated Quantum
Spin Ladders,
A. Honecker, F. Mila and B. Normand,
Phys. Rev. B 94, 094402 (2016).
[3] Efficient Quantum Monte-Carlo Simulations of Highly Frustrated Magnets: the Frustrated
Spin-1/2 Ladder,
S. Wessel, B. Normand, F. Mila and A. Honecker,
SciPost Phys. 3, 005 (2017).
[4] Thermal Critical Points and Quantum Critical End Point in the Frustrated Bilayer Heisenberg
Antiferromagnet,
J. Stapmanns, P. Corboz, F. Mila, A. Honecker, B. Normand and S. Wessel,
Phys. Rev. Lett. 121, 127201 (2018).
[5] Thermodynamic Properties of the Shastry-Sutherland Model from Quantum Monte Carlo Simulations,
S. Wessel, I. Niesen, J. Stapmanns, B. Normand, F. Mila, P. Corboz and A. Honecker,
Phys. Rev. B 98, 174432 (2018).
[6] Thermodynamic Properties of the Shastry-Sutherland Model throughout the Dimer-Product Phase,
A. Wietek, P. Corboz, S. Wessel, B. Normand, F. Mila and A. Honecker,
Phys. Rev. Res. 1, 033038 (2019).
[7] A Quantum Magnetic Analogue to the Critical Point of Water,
J. Larrea Jiménez, S. P. G. Crone, E. Fogh, M. Zayed, R. Lortz, E. Pomjakushina, K. Conder,
A. M. Läuchli, L. Weber, S. Wessel, A. Honecker, B. Normand, Ch. Rüegg, P. Corboz, H. M.
Rønnow and F. Mila,
unpublished (arXiv:2009.14492).