Thermal properties offrustrated quantum magnets
F. MilaEcole Polytechnique Fédérale de Lausanne
Switzerland

Theorists
Stefan Wessel (Aachen)
Andreas Honecker (Cergy)
Bruce Normand (PSI)
Philippe Corboz (Amsterdam)
Olivier Gauthé (Lausanne)

Experimentalists
Henrik Ronnow (Lausanne)
Christian Rüegg (Villigen)and many more…

Scopenn

Thermal properties: quantum Monte Carlo and minus sign
Fully frustrated bilayer:
à Minus sign free QMC in dimer basis
à Ising critical pointn

Shastry-Sutherland model:
à From QMC to iPEPS (tensor network)à SrCu2(BO3)2 under pressure: Critical point

nJ1-J2 model:
à Ising transition revealed by iPEPSnConclusions

Quantum Monte Carlo I
n Partition function as a path integraln Starting point:n Problem: to calculate the exponential ofH, one needs to diagonalize it!

Quantum Monte Carlo II

Quantum Monte Carlo III
No minus sign!

Monte Carlo samplingQuantum Monte Carlo IV
n Heisenberg AF in configuration basis:à All off-diagonal matrix elements are
positive!
à Bipartite lattice: rotation by π on one
sublattice to change the signs of all offdiagonal matrix elements
à Non-bipartite lattice: no way out in
configuration basis

Fully frustrated dimer models
n Example: fully-frustrated ladder

Total spin on a rung is a good quantum number

Hamiltonian in dimer basisn In general, involves both the sum and thedifference of spins on a dimern If all exchange integrals between the spinsof coupled dimers are equal (maximalfrustration), the Hamiltonian can be written
in terms of the sum only
à QMC possible if bipartite lattice of dimersFully frustrated bilayer

Ising critical pointFirst-order transition
Jump in dimer <S.S>
Critical point

Ising 2D: 𝛼 = 0C 𝛼 ln L

Physical realization?

SrCu2(BO3)2
Smith and Keszler, JSSC 1991

Cu2+ -> Spin 1/2
JJD ≈ 85 K

JJ/JD ≈ 0.63

JD

Orthogonal dimer model

Magnetization IAnomalies
n M=0n M=1/8
n M=1/4and many more

From orthogonal dimer toShastry-Sutherland modelFrom FFB to Shastry-Sutherland
Fully-frustrated bilayer

GS = product of dimers

Shastry-Sutherland model

Thermal properties of ShastrySutherland model
Hamiltonian : cannot be writtenin terms of sum of spins of dimers

Up to J/JD = 0.526…
à The model with all off-diagonal matrix elements put
arbitrarily negative has the same GSà <sign>’ à 1 as T à 0
à QMC possible!
nSpecific heat and susceptibility

And above J/JD =0.526?

Tensor networks

Example: Matrix product state in 1D (DMRG)
AB
j1i1

C
j2

i2

j3
i3

Generalization to 2D
PEPS = product of entangled pair states
Verstraete and Cirac, 2004

Variational approach
PEPS: minimize the energy w.r.t. tensor elements
n Other schemes: renormalization (MERA,…)n Advantage: dim=pol(D,N), not exp(N)n Why can it work?
à reproduces the ‘area law’ for the entanglement
entropy in the GS of a local Hamiltoniannn

How large should D be? It depends…

Tensor network for T>0
Purified state: density matrix can be written as
the partial trace of a quantum state in an
enlarged Hilbert space (with extra “ancilla”
degrees of freedom)n T infinite: Singlets between physical and ancilladegrees of freedomn Finite T: imaginary-time evolution from T infinite
nF. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, PRL 2004

From FFB to Shastry-Sutherland
Fully-frustrated bilayer

GS = product of dimers

Shastry-Sutherland modelJ/JD
iPEPS with various setups and bond dimension up to 10

iPEPS for Shastry-Sutherland
Critical point

SrCu2(BO3)2 under pressuren Pressure: expected to change J/JD and foundto increase itn NMR (Waki et al 2007): intermediate phasearound 24 kbar, but 2 Cu sites
à NOT the expected plaquette phase!
n Intermediate phase confirmed by neutron
scattering (Zayed et a, 2017), ESR (Sakuraiet al, 2018), and specific heat (Guo et al,2020)

Intermediate phase
under pressure,but two types ofCu sites

NOT the empty
plaquette phaseSecond derivative of magnetization
Magnetic field response
under pressure

Confirmation of a phasetransition around 2GPaNeutron scattering
Susceptibility

J/JD increases
Full plaquette intermediate phase

ESR
The gap levels offaround 1.8 GPa

Confirmation of a phasetransition around 1.8 GPa

Intermediate phase with criticaltemperature around 2K

iPEPS for Shastry-Sutherland
Critical point around P=19 kbar and T=3.3KCritical point in various models…
Ising in a field

FFB

Shastry-Sutherland… and in water
Tc=374°C
Pc=218 bar

1822: Cagniard
de la Tour

J1-J2 model on square latticeSpin liquid

Chandra and Douçot,PRB 1988
or𝛼 =J2/J1

iPEPS for J1-J2 modeln

Two helical states in collinear phase at large J2
à Ising transition at finite temperature
Chandra, Coleman and Larkin, PRL 1990

nNumerical confirmation?
à QMC: very severe minus sign problem

à iPEPS: Yes if SU(2) symmetry strictly enforced
during imaginary time evolution

J2/J1=0.85Ising transition for J2/J1 large enough from finite T iPEPSPhase diagram of J1-J2 model

Conclusions
Thermal properties of frustrated quantum
magnets are no longer inaccessible
à QMC: sometimes possible, e.g. in dimer basis
à iPEPS: thermal Ising transition and criticalpoint in J1-J2 and Shastry-Sutherland models
n Extensions:à QMC: generalize to other situations
à iPEPS: combine with real-time evolution toaccess T dependence of spectral functionn