# Metastable Kitaev Spin Liquids in Isotropic Quantum Heisenberg Magnets

G. Baskaran  The Institute of Mathematical Sciences, C.I.T. Campus, Chennai
600 113,
Department of Physics, Indian Institute of Technology, Chennai 600 036, India
&
Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada

###### Abstract

Metastable states with surprising properties abound in Hilbert space. We study
unfrustrated isotropic spin-$\rm\frac{1}{2}$ Heisenberg models in honeycomb
lattice and find emergence of metastable Kitaev spin liquids having a 2-spin
nematic long range order, via spontaneous symmetry breaking. Decomposition of
isotropic Heisenberg Hamiltonian $H^{H}$ into an exact sum of 3 noncommuting
(permuted) Kitaev Hamiltonians, $H^{H}$ = $H^{K}_{\rm xyz}+H^{K}_{\rm
yzx}+H^{K}_{\rm zxy},$ helps us build a degenerate manifold of flux free
metastable Kitaev spin liquid vacua and vector Fermionic (Goldstone like)
collective modes. We introduce a method, symmetric decomposition of
Hamiltonians, which might help craft designer metalstable phases. It is likely
that small Kitaev interactions present in Jackeli-Khaliullin-Kitaev materials,
with dominant Heisenberg couplings, bring in metastable Kitaev spin liquid
features in real experiments. Present work opens possibilities of performing
quantum computation and other tasks, using exotic quasiparticles and exotic
metastable states, present in nonexotic real systems.

## I. Introduction

In the incomprehensibly vast many body Hilbert space of model Hamiltonians,
novel metastable states, manybody localization, quantum scars, Hilbert space
fragmentation, time crystals, quantum orders and phenomena, beyond the
familiar ground realities exist [1, 2, 3, 4]. Model Hamiltonians, designed to
understand certain low energy physics and experiments, are space ships that
could take us to new worlds. Such explorations are currently in vogue, thanks
to exciting developments in non-equilibrium quantum manybody physics and
quantum information science.

Equilibrium stable phases with long range order are well known: scores of
crystalline phases of ice, ferro/antiferromagnets, ferro/antiferroelectrics,
superfluids, superconductors etc. Metastable phases, which break translational
invariance, are built using topological defects in underlying Landau type of
order parameters: intertwined network of magnetic domain walls, Skyrmion
glasses in magnets, vortex glasses in type-II superconductors, structural
glasses, defect networks in crystals etc. Degree of metastability, time scale
of relaxation are decided by, among other things, thermal, quantum
fluctuations, interaction between defects etc. There are also defect free
metastable phases, like supercooled water and possibly supersolid He4.

There is growing interest in going beyond Landau’s order parameter paradigm in
quantum matter. For example, quantum spin liquids, present in certain
frustrated Heisenberg spin models [5], anisotropic Kugel-Khomskii-Kitaev
compass models [7, 8] etc., harbor exotic neutral Fermion, anyon excitations,
emergent gauge fields, quantum orders etc [9, 10, 11, 12]. Spin liquids are
sources of rich physics and connect us to worlds beyond condensed matter and
rich mathematics [13, 14].

Metastable quantum phases with quantum orders or symmetry protected
topological orders remain vastly unexplored. We suggest Symmetric
Decomposition of Hamiltonians (SDH), a route to explore or sculpt designer
metastable vacua, having quantum or topological orders. It is illustrated
using unfrustrated isotropic Heisenberg spin models. Isotropic
spin-$\rm\frac{1}{2}$ Heisenberg Hamiltonians with nearest neighbor FM/AFM
coupling in honeycomb lattice are known to have spontaneous long range FM/AFM
ordered ground states. We build, in these unfrustrated isotropic Heisenberg
systems, metastable Kitaev spin liquid vacua, exhibiting spontaneous two spin
nematic order, quantum order, Goldstone like Fermionic collective modes etc.

Decay of the false Kitaev spin liquid vacuum of an isotropic Heisernberg
magnet is a difficult question to analyse; it is beyond the scope of the
present article. However, we hypothesize a three stage decay, with a
superimposed transient and time crystal like local oscillations.

In Jackeli-Khaliullin-Kitaev materials [15], with a dominant Heisernberg
coupling, metastable Kitaev spin liquid phenomena are likely to intervene in
real experiments, because of an enhanced Kitaev spin liquid susceptibiliy
exhibited by isotropic Heisenberg systems. This could explain some of the
anomalies and experimental puzzles.

Our work opens possibilities of performing quantum computation and other
tasks, using exotic quasiparticles and exotic metastable states hiding in real
nonexotic systems.

## Symmetries of Quantum Spin Liquids

Anderson proposed spin singlet short range RVB (quantum spin liquid) ground
state for spin-$\rm\frac{1}{2}$ isotropic Heisenberg antiferromaget [5], on a
frustrated triangular lattice. This paramagnetic quantum spin liquid state has
no long range 1200 type single spin order. Further, phase relations among bond
singlet configurations can be varied suitably to define family of quantum spin
liquids, beyond the simple short range RVB [6], containing rich variety of
quantum orders, topological orders etc.

It also is possible to have stable paramagnetic quantum spin liquid ground
states, which have spontaneous long range two spin nematic orders [16, 17, 18,
19, 20], showing divergence of corresponding 2 spin susceptibilities, obtained
via 4 spin correlation functions. Or three spin scalar chiral orders [21],
exhibiting divergence of corresponding 3 spin susceptibilities, obtained via
suitable 6 spin correlation functions. These generalised RVB states have
resonating singlets and triplets appearing together in a variety of ways and
define a plethora of quantum spin liquids.

Kitaev spin liquid ground state (zero flux sector) is anisotropic in spin
space. It has lesser symmetry, from the point of view of global spin symmetry
present in isotropic Heisenberg Hamiltonains. However, Kitaev spin liquid has
the full symmetry of the Kitaev spin Hamiltonian that has manifest bond
anisotropy. In the present article, we show emergence of such metastable
phases and corresponding Goldstone modes, even in the absence of frustration
in spin-$\rm\frac{1}{2}$ isotropic Heisenberg magnets. (We also find [22] that
we could stablize such a spontaneuos symmetry broken Kitaev spin liquid like
ground state having nematic order, by incorporating non neighbor Heisernberg
couplings in spin-$\rm\frac{1}{2}$ , isotropic Heisenberg Hamiltonian in
honeycomb lattice).

## FM or AFM Order in Isotropic Heisenberg Hamiltonians

To study AFM ordered ground state arising via spontaneous symmetry breaking,
we traditionally decompose isotropic spin Hamiltonians into solvable pieces.
Then start with ground state solution of one of the solvable pieces and
include effects of the rest of the pieces using certain approximations. For
example, we decompose non-frustrated nearest neighbor S=$\rm\frac{1}{2}$
Heisenberg Hamiltonian $H^{H}$ in a bipartite lattice into three noncommuting
Ising pieces,

$H^{H}=J\sum_{\langle
ij\rangle}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}=J\sum_{\langle
ij\rangle}({\sigma}^{x}_{i}{\sigma}^{x}_{j}+{\sigma}^{y}_{i}{\sigma}^{y}_{j}+{\sigma}^{z}_{i}{\sigma}^{z}_{j})$
(1)

and perform spin wave (stability) analysis, about classical (direct product)
ground state of one of the three pieces, say z-axis Ising piece. Holstein-
Primakoff transformation and large S approximation is often to include
remaining XY piece of the Hamiltonian and study effects of residual spin
interactions (transverse quantum fluctuations) and obtain Goldstone modes.

Presence of positive frequency quadratic/linear Goldstone modes for AFM,
indicates emergence of long range order via spontaneous symmetry breaking. In
the case of unfrustrated AFM’s in dimensions 2 and above, transverse zero
point spin fuctuations lead to (finite, non-divergent) zero point reduction in
sublattice magnetization. Quantum fluctuations also modify Ising AFM (product)
state into quantum entangled many body spin states.

## Looking for
Metastable Kitaev Spin Liquids in
Isotropic Heisenberg Hamiltonians

We observe that isotropic SU(2) symmetric Heisenberg Hamiltonian $H^{H}$, with
nearest neighbor $\langle ij\rangle$ coupling in a honeycomb lattice can be
symmetrically decomposed exactly, into three noncommuting pieces of Kitaev
Hamiltonians,

$H^{H}=J\sum_{\langle
ij\rangle}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}~{}\equiv~{}H^{K}_{xyz}+H^{K}_{yzx}+H^{K}_{zxy}$
(2)

where the reference Kitaev piece

$H^{K}_{xyz}\equiv J\sum_{\langle
ij\rangle_{x}}{\sigma}^{x}_{i}{\sigma}^{x}_{j}+J\sum_{\langle
ij\rangle_{y}}{\sigma}^{y}_{i}{\sigma}^{y}_{j}+J\sum_{\langle
ij\rangle_{z}}{\sigma}^{z}_{i}{\sigma}^{z}_{j}$ (3)

The three noncommuting Kitaev pieces are related by cyclic permutations of
spin components along 3 nearest neighbor bonds, denoted as ${\langle
ij\rangle}_{x},{\langle ij\rangle}_{y}$ and ${\langle ij\rangle}_{z}$ of the
reference piece $H^{K}_{xyz}$ (equation 3).

In the supplementary section we summarise exact solution of manybody spectrum
[8] of the reference Kitaev Hamiltonian. Kitaev writes Pauli spin operators
$\sigma^{x}_{i}=ic^{x}_{i}c_{i}$, $\sigma^{y}_{i}=ic^{y}_{i}c_{i}$ and
$\sigma^{z}_{i}=ic^{z}_{i}c_{i}$, using four Majorana Fermion operators
$(c^{x}_{i},c^{y}_{i},c^{z}_{i},c_{i})\equiv({\vec{c}}_{i},c_{i})$ (vector and
a scalar), of Hilbert space dimension 4. This representation preserves
commutation relation between components of the Pauli spin operators. The
condition $\sigma^{x}_{i}\sigma^{y}_{i}\sigma^{z}_{i}=i$ for each site,
imposes a local constraint $c^{x}_{i}c^{y}_{i}c^{z}c_{i}=1$, and reduces
dimension of the 4-dimensional Hilbert space of Majorana Fermion modes into
2-dimensional physical Hilbert space of Pauli spin operators.

In what follows we pick a reference Kitaev Hamiltonian piece $H^{K}_{xyz}(0)$,
in the zero flux sector \- it is the Hamiltonian of free emergent Majorana
Fermions hopping on the honeycomb lattice (equation 16, supplementary
section), in the absence of flux (Bp = 1) in every plaquette. Then we study,
using certain approximations, how this reference metastable Kitaev liquid
vacuum and its low energy excitation spectra are modified by interactions
among Majorana Fermions, arising from interaction terms of remaining two
Kitaev pieces.

As discussed in the next section, in the renormalized vacuum sector, Majorana
Fermion hopping parameter gets modified and new vector Majorana Fermion
collective modes emerge. All excitations have frequencies greater than or
equal to zero. This qualifies the reference zero flux Kitaev spin liquid state
as a metastable or locally stable phase, within our approximations.

## Interactions Modified
Metastable Kitaev spin liquids in
isotropic Heisenberg Magnets

In the Majorana Fermion representation for spin-$\rm\frac{1}{2}$ operators
Heisenberg Hamiltonian has the manifest rotational invariance, with respect to
global SO(3) rotation of the vectors ${\vec{c}}_{i}$’s:

$H=J\sum_{\langle
ij\rangle}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}=J\sum_{\langle
ij\rangle}c_{i}c_{j}({\vec{c}}_{i}\cdot{\vec{c}}_{j})$ (4)

Scalar and vector Majorana Fermions get delocalized over the honeycomb
lattice, via isotropic 2-body interactions. However, this 2-body interaction
prevents exact solvability of SU(2) symmetric Heisenberg Hamiltonian, even in
the enlarged Hilbert space.

With a focuss on building metastable vacuua, we proceed to find the effect of
the two remaining Kitaev pieces on the ground state Kitaev spin liquid sector
of the reference Kitaev piece, using a Hartree type of approximate analysis.
We introduce Heisenberg Hamiltonian HH(0;xyz), for the ground state (zero
flux) sector and take care of quantum fluctuations about the zero flux sector
of reference Kitaev spin liquid (described by $H^{K}_{xyz}(0)$), arising from
interaction among Majorana Fermions, present in the remaining two Kitae pieces
$H^{K}_{yzx}+H^{K}_{zxy}$.

The effective Heisenberg Hamiltonian HH(0;xyz), corresponding to reference
zero flux sector acquires a specific anisotropic two body Majorana Fermion
interaction form:

$H^{H}(0;xyz)=H^{K}_{xyz}(0)+H^{K}_{yzx}+H^{K}_{zxy}~{}~{},$ (5)

where

$\displaystyle H^{K}_{xyz}(0)$ $\displaystyle\equiv$ $\displaystyle
J\sum_{\langle ij\rangle}ic_{i}c_{j}{~{}~{}~{}\rm and}$ (6) $\displaystyle
H^{K}_{yzx}+H^{K}_{zxy}$ $\displaystyle\equiv$ $\displaystyle+J\sum_{\langle
ij\rangle_{x}}c_{i}c_{j}(c^{y}_{i}c^{y}_{j}+c^{z}_{i}c^{z}_{j})+$
$\displaystyle+J\sum_{\langle ij\rangle_{y}}c_{i}c_{j}(c^{z}_{i}c^{z}_{j}$
$\displaystyle+$ $\displaystyle c^{x}_{i}c^{x}_{j})+J\sum_{\langle
ij\rangle_{z}}c_{i}c_{j}(c^{x}_{i}c^{x}_{j}+c^{y}_{i}c^{y}_{j})~{}~{}$ (7)

In HH(0;xyz), corresponding to zero flux sector, scalar Fermion has an
independent dynamics (first term, from reference Kitaev piece). Vector
Fermions have no indpendent dynamics. Scalar Majorana Fermion has bond
dependent 2-body interactions with the vector Fermions (arising from remaining
two Kitaev pieces, equation 7), which help delocalizes vector Fermions in an
anisotropic fashion. This is to be contrasted with the full isotropic
Heisenberg Hamiltonian (equation 4), where scalar and vector Fermions get
delocalized in an isotropic fashion, via 2-body interactions.

Now we perform a uniform Hartree factorization of interaction terms (equation
6) and get Hartree appoximated Heisenberg Hamiltonian ${H}^{H}_{H}(0;xyz)$ in
the zero flux sector:

$\displaystyle{H}^{H}_{H}(0;xyz)=(J+\alpha_{v})\sum_{\langle
ij\rangle}ic_{i}c_{j}$ $\displaystyle+$ (8)
$\displaystyle+(J+\alpha_{s})\\{\sum_{\langle
ij\rangle_{x}}i(c^{y}_{i}c^{y}_{j}+c^{z}_{i}c^{z}_{j})$ $\displaystyle+$
$\displaystyle\sum_{\langle
ij\rangle_{y}}i(c^{z}_{i}c^{z}_{j}+c^{x}_{i}c^{x}_{j})+$
$\displaystyle+\sum_{\langle
ij\rangle_{z}}i(c^{x}_{i}c^{x}_{j}+c^{y}_{i}c^{y}_{j})\\}+{\rm constant,}$ (9)

where the Hartree averages,
$\alpha_{v}\equiv\frac{2}{3}\langle{\vec{c}}_{i}\cdot{\vec{c}}_{j}\rangle$ and
$\alpha_{s}\equiv\langle{c}_{i}c_{j}\rangle$ are determined selfconsistently.
We define renormalized hopping parameters, J${}_{v}\equiv J(1+\alpha_{v})$ for
vector and J${}_{s}\equiv J(1+\alpha_{s})$ for scalar Fermions. After
regrouping, above Hamiltonian takes a more transparent form:

$\displaystyle{H}^{H}_{H}(G,xyz)=J_{s}\sum_{\langle ij\rangle}ic_{i}c_{j}$
$\displaystyle+$ $\displaystyle+J_{v}\\{\sum_{\langle
ij\rangle_{yz}}ic^{x}_{i}c^{x}_{j}+\sum_{\langle
ij\rangle_{zx}}ic^{y}_{i}c^{y}_{j}$ $\displaystyle+$
$\displaystyle\sum_{\langle
ij\rangle_{xy}}ic^{z}_{i}c^{z}_{j}\\},~{}~{}~{}~{}$ (10)

where the new bond symbol $\langle ij\rangle_{xy}$ stands for nearest neighbor
hopping of z-component of vector Majorana Fermions, along a system of parallel
xy zig-zag chains, made of alternating x and y type of bonds. Similarly x and
y components of vector spins hop along yz and zx zig-zag chain systems
respectively. While scalar Fermions continue to hop isotropically in the
honeycomb lattice, vector Fermions acquire anisotropic 1D chain dynamics. That
is, (x,y,z) components of vector Majorana Fermions become one dimensional and
hop independently only along (yz, zx, xy) system of zig-zag chains
respectively.

Using translational invariance and periodic boundary condition, we get Bloch
states for scalar Fermion with 2D (graphene like) Kitaev spectrum and 1D
spectrum for each components of vector Fermions, which hop along yz, zx and xy
zig-zag chains. All energy eigen values are $\geq$ 0\. This ensures that
Kitaev spin liquid state is not completely destabilized by transverse quantum
fluctuations (present in the isotropic Heisenberg Hamiltonian) and survives as
a metastable one. Scalar Fermion spectrum is renormalized. While the scalar
Majorana Fermions have Dirac nodes at K and K’ points in, three components of
vector Fermions have three independent sets of zero energy lines in the BZ. We
have Majorana Fermi surfaces in the 2D BZ, somewhat similar to Kitaev model in
the Fisher lattice [25].

## Spontaneous 2 spin Nematic Metastable order in Isotropic Heisenberg Magnets

Metastable Kitaev spin liquid phase is anisotropic, as evidenced from 1D
nature of the three components of vector Majorana Fermions, delocalized
independently along three different sets of zig-zag chains. There is a nematic
spin liquid order arising via symmetry breaking (from choice of reference
Kitaev piece) in an SU(2) symmetric Heisenberg spin system. Vector Fermions,
which were organized as static Z2 gauge field in Kitaev model, organize
themselves as positive frequency vector Fermion collective modes, in our
isotropic Heisenberg Hamiltonian.

We obtain a manifold of Kitaev spin liquid metastable vacuua generated by
SO(3) rotation. This is achieved by global rotation of spin axes of the
reference Kitaev piece. The vector Fermionic modes we have obtained are
collective modes, which represent generalized Goldstone modes. Interestingly,
the Kitaev spin liquid metastable vacua exist for both Ferro or
Antiferromagnetic isotropic Heisenberg systems (for J positive and negative).

Kitaev Hamiltonian is known to have highly anisotropic and ultra short range
two spin correlation functions [23], consistent with the anisotropic bond
dependent Ising spin-spin interaction in Kitaev spin Hamiltonian. In
particular,

$\displaystyle\langle\sigma_{i}^{\alpha}\sigma_{j}^{\beta}\rangle=g_{\langle
ij\rangle_{\alpha}}\delta_{\alpha\beta}~{}~{}~{}~{}{\rm
i,j~{}nearest~{}neighbors}$
$\displaystyle{\rm~{}~{}~{}~{}~{}~{}~{}~{}~{}=~{}0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}otherwise}$
(11)

In our approximation we find that bond dependent anisotropy gets modified only
quantitatively, by quantum fluctuations contained in equation 7. An accurate
and quantitative estimate of correction to spin-spin correlation function
(unlike the case of pure Kitaev model [23]) involves projection of the
Majorana Fermion wave function into physical spin Hilbert space, which we are
not performing in the present article.

From the point of view of full rotational spin symmetry of isotropic
Heisenberg Hamiltonian, above anisotropic spin-spin correlations represent
emergent two spin nematic order present in the metastable Kitaev spin liquid
we have constructed. It is a three sublattice (3 types of nearest neighbor
bonds) nematic order (represented by three directors, headless vectors), lying
parallel to x,y and z directions in spin space.

It is also important to remember that eventhough we know approximate manybody
spectrum, the manybody wave functions, including the ground state is not known
in the physical Hilbert space, interms of physical spin variables.

## Crafting by Symmetric Decomposition of Hamiltonians

Often our focuss is on low energy manybody spectrum and low energy physics of
model Hamiltonians. However, these Hamiltonians also organize a rich spectrum
of excited manybody states and metastable phases in the Hilbert space. We can
get information about metastable phases, when models are completely integrable
or use exact diagonalization study of small systems, quantum Monte Carlo
simulations, time dependent studies etc. One could also use renormalization
group study, in multiparameter space of Hamiltonians, to explore metastable
phases by a study of stable and unstable fixed points and flow in the
parameter space.

We saw that symmetric splitting of unfrustrated isotropic Heisenberg
Hamiltonians on the honeycomb lattice, into three noncommuting bond
anisotropic (frustrated) Kitaev spin Hamiltonians (frustration arising from
bond dependent anisotropic spin-spin interactions), helped us to find Kitaev
spin liquid like metastable phases and associated spontaneous nematic order
etc. This suggests an alternative approach symmetric decomposition of
Hamiltonian, using symmetries of subgroups of full lattice translational and
spin rotational symmetries.

Our hypothesis is that residual symmetries in the decomposed Hamiltonian
protects corresponding metastable phases. Further, if the decomposed reference
Hamiltonian piece is frustrated enough (like in Kitaev Hamiltonian) in spin
space, it also ensures spin liquid metastable states.

There are multiple ways for Hamiltonian decomposition, where the pieces have
symmetries of subgroups of lattice translation and spin rotation group. To
illustrate this we introduce complementary Kitaev model $H^{cK}_{xyz}\equiv
H^{K}_{yzx}+H^{K}_{zxy}$, such that isotropic Heisenberg Hamiltonian, is a sum
of Kitaev nd complementary Kitaev Hamiltonians: $H^{H}\equiv
H^{K}_{xyz}+H^{cK}_{xyz}$. We wish to find if metastable vacuua (complementary
Kitaev spin liquid) corresponding to complementary Kitaev model exists. Since
complementary Kitaev model is not exactly solvable, answering this question
becomes difficult.

Let us consider a simpler situation of splitting unfrustrated isotropic
spin-$\rm\frac{1}{2}$ 2D Antiferromagnetic Heisenberg Hamiltonian on a
triangular lattice, into Ising and XY pieces. We ask, is there is a manifold
of metastable states where we have spontaneous Kosterlitz-Thouless XY order
and corresponding Goldstone modes ?

Similarly, spin-$\rm\frac{1}{2}$ XY model Hamiltonian in a square lattice HXY
can be symmetrically decomposed into noncommuting sum of two Kugel-Khomskii
[7] pieces $H^{KK}_{xy}{\rm and}H^{KK}_{yx}$:

$H^{XY}=J\sum_{\langle
ij\rangle}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j})~{}\equiv~{}H^{KK}_{xy}+H^{KK}_{yx}$
(12)

Approximate analysis can be performed to find if square lattice XY FM or AFM
contains metastable anisotropic spin liquid vacuua corresponding to Kuge-
Khomskii compass Hamiltonians.

There is a hope that designer metastable phases could be found out using our
Symmetric Decomposition of Hamiltonians.

Using a similar approach, by adding and substracting SU(2) symmetric,
isotropic spin liquid stabilizing frustrating terms in the frustration free
isotropic Hamiltonians, we could explore RVB type of SU(2) symmetric quantum
spin liquids, which have a rich variety of quantum orders and chirality
orders.

FM/AFM Hamiltonians of other lattices, where Kitaev model can be defined and
exactly solved are being investigated by us [24] looking for metastable Kitaev
spin liquids: Fisher lattice (having Majorana FS [25]), decorated honey comb
lattice exhibiting spontaneous Chiral symmetry breaking [26], non-Archemedian
lattice [27], classical spin Kitaev model [28] 3D Kitaev model [29]. There are
indications that because the lattices are more complex than honeycomb lattice,
Kitaev spin liquids have deeper metastable minimum.

## Preparation of False Vacuum and watching its evolution

Fate of the false vacuum [30] is an important issue, from cosmology to our
present problem of quantum tunneling and decay of metastable Kitaev spin
liquid vacuum of an isotropic Heisenberg spin system. Since we have Jackeli-
Khaliullin-Kitaev Heisernberg [15] materials with dominant isotropic Ferro/AFM
interactions, it will be wonderful to perform real experiments. Also try to
understand existing experimental results and anomalies [31, 32] in the light
of intervension of metastable Kitaev spin liquid phases and quasiparticles.

As a number of quantum processors have emerged and experimental simulations
[33] and computer simulations [34] are being performed on certain manybody
lattice Hamiltonians, we suggest the following protocol to address our
problem: i) prepare, best possible Kitaev spin liquid ground state for a
finite number of qubits, using a reference (FM/AFM) Kitaev spin Hamiltonian
and ii) use the full isotropic spin-half (FM/AFM) Hesenberg Hamiltonian and
study time evolution of the prepared Kitaev spin liquid with a nematic order.

What do we expect ? Our hypothesis is presence of three stages in time
evolution. Since there is a finite energy gap for Z2 flux excitaions, we
expect, first, renormalization of the Kitaev spin liquid state (analogue of
growth of zero point spin fluctuations in an Ising AFM vacuum in 3 or 2D).
After a time interval, the system will nucleate Z2 fluxes, create a small
density of Majorana Fermions as well as topological defects in the Kitaev
nematic order, via quantum tunneling. Finally there will be a crossover to
nucleation of real (FM/AFM) ground state bubbles and their growth.

In a model study we find [35] a remarkable time crystal like local transient
behaviour, a periodic oscillation of the state vector. In short, there may be
many surprises in store, in real experiments and theoretical studies.

## Discussion

In the present article we have explored possibility of metastable spin liquid
states in istropic spin-$\rm\frac{1}{2}$ Heisenberg systems on a honeycomb
lattice. In addition, we have suggested general ways to explore presence of
metastable spin liquid states. While some of the metastable states may turn
out to be locally stable in all directions, others may not be locally stable
in all directions.

Our study also raises some interesting questions. If we get Kitaev spin liquid
and spin nematics in isotropic systems, it allows for topological
singularities in the spin nematic phase, in addition to plaquette flux
excitations. It is likely these topological defects trap non-Abelian anyons as
zero modes.

Taking advantage of known exact specrum of Kitaev honeycomb lattice
Hamiltonian, we have restricted our search to study quantum spin liquids that
do not have manifest global SU(2) symmetry. We also find, in our approximate
study of isotropic Heisenberg magnets, presence of SU(2) symmetric metastable
spin liquids, a la’ RVB states. Under certain conditions time crystal like
transient and local metastable phases emerge.

It is indeed exciting that beyond the comfort zone of equilibrium quantum
statistical mechanics, rich new worlds await even in energy eigen states and
near eigen (manybody wave packet) states of familiar and well studied
Hamiltonians. From the point of view of performing quantum computation and
related tasks, present work opens new avenues to explore and use exotic
quasiparticles and exotic metastable states, which are hiding in nonexotic
real systems.

## Acknowledgement

I thank Francesco Ferrari, R. Shankar and S. Mandal for discussions; and thank
Tony Leggett for his encouraging remarks. Hospitality and stimulating visiting
positions, at The Institute of Mathematical Sciences, Chennai, Indian
Institute of Technology, Chennai and Perimeter Institute for Theoretical
Physics (PI), Waterloo, Canada are acknowledged. PI is supported by the
Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Research and Innovation.

## Supplementary section

## Exact Solution of Kitaev honeycomb lattice model - a Summary

Since we are looking for metastable Kitaev spin liquid phases in isotropic
Heisenberg Hamiltonians, we review Kitaev’s method of finding exact manybody
spectrum of the Kitaev anisotropic spin Hamiltonian in the honeycomb lattce:

$H^{K}_{xyz}\equiv J\sum_{\langle
ij\rangle_{x}}{\sigma}^{x}_{i}{\sigma}^{x}_{j}+J\sum_{\langle
ij\rangle_{y}}{\sigma}^{y}_{i}{\sigma}^{y}_{j}+J\sum_{\langle
ij\rangle_{z}}{\sigma}^{z}_{i}{\sigma}^{z}_{j}$ (13)

Here 3 nearest neighbor bonds, where two spin Ising ineraction term, xx or yy
or zz alone are present are denoted respectively by ${\langle
ij\rangle}_{x},{\langle ij\rangle}_{y}$ and ${\langle ij\rangle}_{z}$.

We assume periodic boundary condition for the hexagonal lattice, having N unit
cells and 2N sites. Each site carries a spin-$\rm\frac{1}{2}$ moment. Kitaev
found that plaquette spin operators (sites 1 to 6 denote nearest neighbor
sites of a given plaquette):

$B_{p}\equiv{\sigma}^{z}_{1}{\sigma}^{x}_{2}{\sigma}^{y}_{3}{\sigma}^{z}_{4}{\sigma}^{x}_{5}{\sigma}^{y}_{6}~{}~{}~{}~{},$
(14)

defined for each elementary plaquette commute among themlselves and with
$H^{K}_{xyz}$. Since B${}^{2}_{p}=1$, eigen values of B${}_{p}=\pm 1$.
Consequently, energy eigenstates of Kitaev Hamiltonian get classified interms
of eigen values of the set $\\{B_{p}\\}$.

To get the full manybody spectrum Kitaev introduces a Majorana Fermion
representation for Pauli spin operators in an enlarged Hilbert space. Each
site has four Majorana Fermion operators,
$(c^{x}_{i},c^{y}_{i},c^{z}_{i},c_{i})\equiv({\vec{c}}_{i},c_{i})$ (vector and
a scalar), of dimension 4. Dimension of Majorana Fermion Hilbert space is 42N.
Dimension of physical spin Hilbert space is 22N.

The representation, $\sigma^{x}_{i}=ic^{x}_{i}c_{i}$,
$\sigma^{y}_{i}=ic^{y}_{i}c_{i}$ and $\sigma^{z}_{i}=ic^{z}_{i}c_{i}$,
preserves commutation relation between components of the Pauli spin operators.
The condition $\sigma^{x}_{i}\sigma^{y}_{i}\sigma^{z}_{i}=i$ for each site,
imposes a local constraint $c^{x}_{i}c^{y}_{i}c^{z}c_{i}=1$, and reduces
dimension of the 4-dimensional Hilbert space of Majorana Fermion modes into
2-dimensional physical Hilbert space of Pauli spin operators.

A specially designed anisotropic structure of the Kitaev Hamiltonian
H${}^{K}_{xyz}$ makes it exactly solvable. In a remarkable fasion, certain
pairs of vector Fermion components appearing in H${}^{K}_{xyz}$ organize
themselves into static and commuting Hartree vector potential, which are
emergent static Z2 gauge fields. One obtains free (noninteracting) scalar
Fermions, which hop and delocalize, in the presence of emergent static Z2
gauge fields:

$\displaystyle H^{K}_{xyz}=J\sum_{\langle
ij\rangle_{x}}c_{i}c_{j}c^{x}_{i}c^{x}_{j}+J\sum_{\langle
ij\rangle_{y}}c_{i}c_{j}c^{y}_{i}c^{y}_{j}+J\sum_{\langle
ij\rangle_{z}}c_{i}c_{j}c^{z}_{i}c^{z}_{j}$ $\displaystyle\equiv
J\sum_{\langle ij\rangle_{x}}ic_{i}c_{j}{\hat{u}^{x}}_{\langle
ij\rangle_{x}}+J\sum_{\langle ij\rangle_{y}}ic_{i}c_{j}{\hat{u}^{y}}_{\langle
ij\rangle_{y}}+J\sum_{\langle ij\rangle_{z}}ic_{i}c_{j}{\hat{u}^{z}}_{\langle
ij\rangle_{z}}$ (15)

Operators ${\hat{u}}^{\alpha}_{\langle ij\rangle_{\alpha}}\equiv-
ic^{\alpha}_{i}c^{\alpha}_{j}$ ($\alpha$ = x,y,z) appearing above commute
among themselves and with H${}^{K}_{xyz}$: consequently they can be treated as
c-numbers in solving the manybody Hamiltonian. Further,
$({\hat{u}}^{\alpha}_{\langle ij\rangle_{\alpha}})^{2}=1$ implies that
${\hat{u}}^{\alpha}_{\langle ij\rangle_{\alpha}}$ = $\pm 1$.

The transformation ${\hat{u}}^{\alpha}_{\langle
ij\rangle_{\alpha}}\rightarrow\tau_{i}{\hat{u}}^{\alpha}_{\langle
ij\rangle_{\alpha}}\tau_{j}$ and $c_{i}\rightarrow\tau_{i}c_{i}$ leaves Kitaev
Hamiltonan invariant. Here $\tau_{i}\pm 1$. We have an emergent local Z2 gauge
invariance and u’s become emergent Z2 gauge fields: Consequently, enlarged
Hilbert space of dimension 42N breaks into 22N gauge copies, each copy having
identical energy spectrum of 22N manybody energy eigen states.

Formally, the exact many body spectrum becomes that of emergent non-
interacting Majorana Fermions hopping in the lattice in the presence of static
${\hat{u}}^{\alpha}_{\langle ij\rangle_{\alpha}}$ fields. Number of distinct
Z2 gauge flux configurations is 2N, where N is the number of elementary
hexagons. Wilson loops that are gauge invariant operators on hexagens, measure
gauge fluxes (0 or $\pi$), get identified with the Bp operators (equation 3).
That is, B${}_{p}\equiv{\hat{u}}^{y}_{\langle
12\rangle_{y}}{\hat{u}}^{z}_{\langle 23\rangle_{z}}{\hat{u}}^{x}_{\langle
34\rangle_{x}}{\hat{u}}^{y}_{\langle 45\rangle_{y}}{\hat{u}}^{z}_{\langle
56\rangle_{z}}{\hat{u}}^{x}_{\langle 61\rangle_{x}}$. Eigen values $\pm 1$ of
Bp corresponds to 0 and $\pi$ fluxes of emergent Z2 gauge fields.

It follows from Lieb theorem that ground state of Kitaev Hamiltonian is the
lowest energy state of zero Z2 flux, $\\{B_{p}=1\\}$ sector. Hamiltonian
H${}^{K}_{xyz}(0)$ of the zero flux sector is obtained by putting all
${\hat{u}}^{\alpha}_{\langle ij\rangle_{\alpha}}=1$ in equation (4):

$H^{K}_{xyz}(0)=J\sum_{\langle ij\rangle}ic_{i}c_{j}$ (16)

That is, in the zero flux sector, free scalar Majorana fermions hop
isotropically in the honeycomb lattice. In this sector the vector Majorana
Fermion degree of freedom essentially disappear from the problem.

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