# Presentation: 681711

## Quasiparticle spectrum termination in quantum spin liquid

- Igor Zaliznyak
*Neutron Scattering Group, Brookhaven National Laboratory*

## Quasiparticles

**1 meV = 11.6 K**

## Neutron scattering: how neutrons measure excitations.

## How neutrons measure excitations now.

*B. Brokhouse (1961) *

*I. A. Zaliznyak and S.-H. Lee, *
*in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005) *

## What is quantum liquid?

-  What is quantum liquid?
-  all of the above at *T* → 0 (i.e. at temperatures much lower than inter-particle interactions in the system)

-  Elemental quantum liquids:
-  H, He and their isotopes
-  made of light atoms  strong quantum fluctuations

## Excitations in quantum Bose liquid: superfluid 4He

- phonon

- roton

- maxon

- Excitations in quantum Bose liquid: superfluid 4He

*Woods & Cowley, Rep. Prog. Phys. 36 (1973)*

## The “cutoff point” of the quasiparticle spectrum in the quantum Bose-liquid

## Spectrum termination in 4He: experiments

**H** = **q**ε (q) *a***q**+ *a***q** + **q**,**q**′ V**q**,**q**′(*a***q***a*+**q**′*a*+**q**-**q**′ + H.c.) + ...

## Spectrum termination in 4He: experiment

## What is quantum spin liquid?

-  Quantum liquid state for a system of Heisenberg spins
**H** = J|| ***S****i****S****i*+||+ J***S****i****S****i*

-  Exchange couplings J||, J through orbital overlaps may be different
-  J||/J >> 1 (<<1) parameterize quasi-1D (quasi-2D) case

## Simple example: coupled S=1/2 dimers

***H*** = *J*0 ***S****1****S****2**J*0/2 (***S****1** + ****S****2*)2 + *const.  *

- Single dimer: antiferromagnetically coupled S=1/2 pair

## Simple example: coupled S=1/2 dimers

***H*** = *J*0 ***S****2i****S****2i+1**J*1 (***S****2i** ****S****2i+2*)

- Chain of weakly coupled dimers

- Dispersion ***(q) ~ J**0** + J**1**cos(q)*

- triplet

- 

## Dimers in 1D (aka alternating chain)

- Chains of weakly interacting dimers in
- Cu(NO3)2x2.5D2O

**Cu****2+**** ******** ****3d****9**** **
**S=1/2**

- E (meV)

## Weakly interacting dimers in Cu(NO3)2x2.5D2O

*D. A. Tennant, C. Broholm, et. al. PRB 67, 054414 (2003)*

*Spin excitations never cross into 2-particle continuum and live happily ever after*

## What would be a “spin solid”?

-  Heisenberg antiferromagnet with classical spins, **S >> 1**

## 1D quantum spin liquid: Haldane spin chain

-  Heisenberg antiferromagnetic chain with **S = 1**

## Spin-quasiparticles in Haldane chains in CsNiCl3

## Spin-quasiparticles in Haldane chains in CsNiCl3

- Spin-quasiparticles in Haldane chains in CsNiCl3

## Spectrum termination point in CsNiCl3

*I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001) *

## 2D quantum spin liquid: a lattice of frustrated dimers

- M. B. Stone, I. Zaliznyak, et. al. PRB (2001)
- (C4H12N2)Cu2Cl6 (PHCC)

-  singlet disordered ground state
-  gapped triplet spin excitation

**Cu****2+**** ******** ****3d****9**** **
**S=1/2**

## PHCC: a two-dimensional quantum spin liquid

- gap  = 1 meV
- bandwidth = 1.8 meV

## Quasiparticle spectrum termination line in PHCC

- max{E2-particle (q)}

- min{E2-particle (q)}

- E1-particle(q)

- Spectrum termination line

## PHCC: dispersion along the diagonal

-  Q = (0.5,0,-1.5)

-  resolution-corrected fit

-  Q = (0.25,0,-1.25)

- resolution-corrected fit

-  Q = (0.15,0,-1.15)

-  resolution-corrected fit

- Intensity (counts in 1 min)

- E (meV)

- E (meV)

## 2D map of the spectrum along both directions

## Compare: spectrum end point in helium-4

## Summary and conclusions

- Quasiparticle spectrum termination at E > 2 is a generic property of quantum Bose (spin) fluids
- observed in the superfluid 4He
- observed in the Haldane spin chains in CsNiCl3
- observed in the 2D frustrated quantum spin liquid in PHCC
- A real physical alternative to the *ad-hoc* “excitation fractionalization” explanation of scattering continua
- Implications for the high-Tc cuprates: spin gap implies disappearance of coherent spin modes at high E

## Thanks to collaborators

- M. B. Stone
- C. Broholm, D. Reich, T. Hong
- S.-H. Lee
- S. V. Petrov

## Temperature dependence in PHCC

## Temperature dependence in copper nitrate

## Dispersion along the side (l) in PHCC

- Intensity (counts in 1 min)

- E (meV)

## Neutron scattering cross-section

## Quasiparticle cross-section

## How do neutrons measure quasiparticles.

*B. Brokhouse (1961) *