Spin liquid state in

What's a quantum spin liquid?

Conventional magnets

Order at low temperatures by breaking symmetry, classical product states, short-range entangled

Quantum spin liquids

Long-range entanglement, do not order due to quantum fluctuations, non-local fractional excitations (spinons)

broken symmetry $\rightarrow$ collective excitations (magnons)

What's a quantum spin liquid?

Mechanisms:

Example: triangular lattice antiferromagnet

ordered state

+

+ $\dots$

spin liquid

low dimensionality

frustration

Quantum spin liquid in $\text{Ba}_4\text{Ir}_3\text{O}_{10}$ ?

inverse susceptibility

heat capacity

Insulator with T-linear heat capacity
Interactions ~ 500 K but orders at 0.2 K
2D but not geometrically frustrated

Different descriptions

Electron model:

$H = t_{ij}c_i^{\dagger}c_j +v_{ijkl}c_i^{\dagger}c_j^{\dagger}c_lc_k + \dots$

Spin model:

$H = J_{ij}\mathbf{S}_i.\mathbf{S}_j + J_{ijkl}(\mathbf{S}_i.\mathbf{S}_j)(\mathbf{S}_k.\mathbf{S}_l) + \dots$

Parton theory:

$H = t_{ij}f_i^{\dagger}A_{ij}f_j + \dots$

review article: Quantum Spin Liquids, Savary and Balents, 2016

Relevant interactions

$\text{Ir}^{4+}:\ 5d^5$ valence electrons
crystal field ~ on-site Coulomb repulsion > Hund's coupling $\geq$ spin-orbit coupling $\geq$ hopping

Pseudo spin-halfs (?)

Ab initio wave function calculations

Face shared octahedra

Calculating valence electron wave functions for embedded clusters including all relevant interactions

Low lying energy levels:

Face shared

Corner shared

Effective hamiltonian: electrons $\rightarrow$ spins

H = -t\sum_{\langle i, j\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma} + U\sum_{i}n_{i\uparrow}n_{i\downarrow}

H = -t\sum_{\langle i, j\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma} + U\sum_{i}n_{i\uparrow}n_{i\downarrow}

Hubbard model:

Strong coupling limit: $U \gg t$

Integrate out high energy doublons

Exchange process

H_{\text{eff}} = J\mathbf{S}_1.\mathbf{S}_2

H_{\text{eff}} = J\mathbf{S}_1.\mathbf{S}_2

J \sim t^2/U

J \sim t^2/U

At quartic order: next-nearest neighbors, ring exchange ( $\sim t^4/U^3$ )

Exchange with SOC

H = -t\sum_{\langle i, j\rangle,\sigma,\sigma'}\tilde{a}_{i\sigma}^{\dagger}\tilde{a}_{j\sigma'} + U\sum_{i}n_{i\uparrow}n_{i\downarrow}

H = -t\sum_{\langle i, j\rangle,\sigma,\sigma'}\tilde{a}_{i\sigma}^{\dagger}\tilde{a}_{j\sigma'} + U\sum_{i}n_{i\uparrow}n_{i\downarrow}

Pseudospin flips allowed!

H_{\text{eff}}= \mathbf{S}_1.R^{T}JR.\mathbf{S}_2

H_{\text{eff}}= \mathbf{S}_1.R^{T}JR.\mathbf{S}_2

=J\cos\phi\mathbf{S}_1.\mathbf{S}_2 + (1-\cos\phi)(\hat{\mathbf{d}}.\mathbf{S}_1)(\hat{\mathbf{d}}.\mathbf{S}_2) + \sin\phi\hat{\mathbf{d}}.(\mathbf{S}_1\times\mathbf{S}_2)

=J\cos\phi\mathbf{S}_1.\mathbf{S}_2 + (1-\cos\phi)(\hat{\mathbf{d}}.\mathbf{S}_1)(\hat{\mathbf{d}}.\mathbf{S}_2) + \sin\phi\hat{\mathbf{d}}.(\mathbf{S}_1\times\mathbf{S}_2)

Frustration on a square

with second order interactions?

Two-site spectrum identical to Heisenberg

Partons

\mathbf{S}_i=\dfrac{1}{2}f_{i\alpha}^{\dagger}\mathbf{\sigma}_{\alpha\beta}f_{i\beta}

\mathbf{S}_i=\dfrac{1}{2}f_{i\alpha}^{\dagger}\mathbf{\sigma}_{\alpha\beta}f_{i\beta}

f_{i\alpha}^{\dagger}f_{i\alpha} = 1

f_{i\alpha}^{\dagger}f_{i\alpha} = 1

$f_i$ are spin-1/2 fermions (spinons)

constraint: 1 fermion per site (gauge invariance, $f_{i\alpha}\rightarrow e^{i\Lambda_i}f_{i\alpha}, A_{ij} \rightarrow A_{ij}+\Lambda_i - \Lambda_j$ )

\mathbf{S}_i.\mathbf{S}_j=-\dfrac{1}{2}f_{i\alpha}^{\dagger}f_{j\alpha}f^{\dagger}_{j\beta}f_{i\beta}-\dfrac{1}{4}f_{i\alpha}^{\dagger}f_{i\alpha}f^{\dagger}_{j\beta}f_{j\beta}

\mathbf{S}_i.\mathbf{S}_j=-\dfrac{1}{2}f_{i\alpha}^{\dagger}f_{j\alpha}f^{\dagger}_{j\beta}f_{i\beta}-\dfrac{1}{4}f_{i\alpha}^{\dagger}f_{i\alpha}f^{\dagger}_{j\beta}f_{j\beta}

Interacting spinons: decouple at mean field level, add gauge field fluctuations

Non-local fractional excitations:

Local excitations have to create spinons in pairs to be gauge invariant: $f_i^{\dagger}$ is not gauge invariant, $f_i^{\dagger}f_i$ is

$f_{1\alpha}^{\dagger}\exp(iA_{12} + iA_{23} +\dots + iA_{(n-1)n})f_{n\alpha}$ is also gauge invariant

Depending on the phase of the gauge field, spinons can be deconfined and are nonlocal

confined in AF

deconfined in QSL

Gutzwiller projection

Instead of gauge fields, project out doublons numerically (with Monte Carlo) from free states

Different noninteracting $|\Psi_0\rangle$ lead to different QSL

Possible in $\text{Ba}_4\text{Ir}_3\text{O}_{10}$ : $U(1)$ QSL with $|\psi_0\rangle$ a metallic free fermion state

spinon

Fermi surface with $T$ -linear heat capacity

Quantum spin liquid in Ba4Ir3O10\text{Ba}_4\text{Ir}_3\text{O}_{10}Ba4​Ir3​O10​?

Quantum spin liquid in $\text{Ba}_4\text{Ir}_3\text{O}_{10}$ ?