We present data on the optical conductivity of URu$_{2-x}$(Fe,Os)$_{x}$Si$_{2}$. While the parent material URu$_2$Si$_2$ enters the enigmatic hidden order phase below 17.5 K, an antiferromagnetic phase is induced by the substitution of Fe or Os onto the Ru sites. We find that both the HO and the AFM phases exhibit an identical gap structure that is characterized by a loss of conductivity below the gap energy with spectral weight transferred to a narrow frequency region just above the gap, the typical optical signature of a density wave. The AFM phase is marked by strong increases in both transition temperature and the energy of the gap associated with the transition. In the normal phase just above the transition the optical scattering rate varies as $ω^2$. We find that in both the HO and the AFM phases, our data are consistent with elastic resonant scattering of a Fermi liquid. This indicates that the appearance of a coherent state is a necessary condition for either ordered phase to emerge. Our measurements favor models in which the HO and the AFM phases are driven by the common physics of a nesting-induced density-wave-gap.
The onset of antiferromagnetic order in URu2Si2 has been studied via neutron diffraction in a helium pressure medium, which most closely approximates hydrostatic conditions. The antiferromagnetic critical pressure is 0.80 GPa, considerably higher than values previously reported. Complementary electrical resistivity measurements imply that the hidden order-antiferromagnetic bicritical point far exceeds 1.02 GPa. Moreover, the redefined pressure-temperature phase diagram suggests that the superconducting and antiferromagnetic phase boundaries actually meet at a common critical pressure at zero t…
We have investigated the nature of the antiferromagnetic (AF) phase induced by uniaxial stress sigma in URu2Si2, by performing elastic neutron scattering measurements up to 0.4 GPa. We have found that the AF Bragg-peak intensity shows a clear hysteresis loop with sigma under the zero-stress cooling condition. The result strongly suggests that the sigma-induced AF phase is metastable and separated from the coexisting "hidden ordered" phase by a first-order phase transition. We also present the analyses of the crystalline strain effects, and suggest that the c/a ratio plays an important role in …
We have investigated the nature of the competition between hidden order and antiferromagnetic (AF) order in URu_2Si_2 by performing the neutron scattering experiments under hydrostatic-pressure P, uniaxial-stress sigma, and Rh-substitution conditions. Hidden order observed at ambient pressure in pure URu_2Si_2 is found to be replaced by the AF order by applying P, sigma along the tetragonal basal plane, and by doping Rh. We discuss these experimental results on the basis of the crystalline strain calculations, and suggest that this phase transition is generated by the 0.1% increase of the tetr…
Hidden-order phase transition in the heavy-fermion superconductor URu$_2$Si$_2$ exhibits the mean-field-like anomaly in temperature dependence of heat capacity. Motivated by this observation, here we explore the impact of the complex order parameter fluctuations on the thermodynamic properties of the hidden order phase. Specifically, we employ the mean-field theory for the hidden order which describes the hidden order parameter by an average of the hexadecapole operator. We compute the gaussian fluctuation corrections to the mean-field theory equations including both the fluctuations due to 'h…
We report a neutron scattering study of the magnetic excitation spectrum in each of the three temperature and pressure driven phases of URu$_2$Si$_2$. We find qualitatively similar excitations throughout the (H0L) scattering plane in the hidden order and large moment phases, with no changes in the $\hbarω$-widths of the excitations at the $Σ$ = (1.407,0,0) and $Z$ = (1,0,0) points, within our experimental resolution. There is, however, an increase in the gap at the $Σ$ point from 4.2(2) meV to 5.5(3) meV, consistent with other indicators of enhanced antiferromagnetism under pressure.
We use neutron scattering to compare the magnetic excitations in the hidden order (HO) and antiferromagnetic (AFM) phases in URu2-xFexSi2 as a function of Fe concentration. The magnetic excitation spectra change significantly between x = 0.05 and x = 0.10, following the enhancement of the AFM ordered moment, in good analogy to the behavior of the parent compound under applied pressure. Prominent lattice-commensurate low-energy excitations characteristic of the HO phase vanish in the AFM phase. The magnetic scattering is dominated by strong excitations along the Brillouin zone edges, underscori…
We have performed elastic neutron-scattering experiments on the heavy-electron compound URu_2Si_2 for pressure P up to 2.8 GPa. We have found that the antiferrmagnetic (100) Bragg reflection below T_m ~ 17.5 K is strongly enhanced by applying pressure. For P < 1.1 GPa, the staggered moment mu_o at 1.4 K increases linearly from ~ 0.017(3) mu_B to ~ 0.25(2) mu_B, while T_m increases slightly at a rate ~ 1 K/GPa, roughly following the transition temperature T_o determined from macroscopic anomalies. We have also observed a sharp phase transition at P_c ~ 1.5 GPa, above which a 3D-Ising type of an…
We have performed the elastic neutron scattering experiments under uniaxial stress σalong the tetragonal [100], [110] and [001] directions for URu2Si2. For σ// [100] and [110], the antiferromagnetic moment μ_o is strongly enhanced from 0.02 μ_B (σ=0) to 0.22 μ_B (σ=2.5 kbar) at 1.5 K. The rate of increase dμ_o/dσis roughly estimated to be ~ 0.1 μ_B/kbar, which is much larger than that for the hydrostatic pressure (~ 0.025 μ_B/kbar). Above 2.5 kbar, μ_o shows a tendency to saturate similar to the behavior in the hydrostatic pressure. For σ// [001], on the other hand, μ_o shows only a slight inc…
The hidden order phase in URu$_2$Si$_2$ is highly sensitive to electronic doping. A special interest in silicon-to-phosphorus substitution is due to the fact that it may allow one, in part, to isolate the effects of tuning the chemical potential from the complexity of the correlated $f$ and $d$ electronic states. We investigate the new antiferromagnetic phase that is induced in URu$_2$Si$_{2-x}$P$_x$ at $x\gtrsim0.27$. Time-of-flight neutron diffraction of a single crystal ($x=0.28$) reveals $c$-axis collinear $\mathbf{q}_\mathrm{m}=(\frac12,\frac12,\frac12)$ magnetic structure with localized …
Simultaneous neutron-scattering and thermal expansion measurements on the heavy-fermion superconductor URu2Si2 under hydrostatic pressure of 0.67 GPa have been performed in order to detect the successive paramagnetic, hidden order, and large moment antiferromagnetic phases on cooling. The temperature dependence of the sharp low energy excitation at the wavevector Q_0=(100) shows that this excitation is clearly a signature of the hidden order state. In the antiferromagnetic phase, this collective mode disappears. The higher energy excitation at the incommensurate wavevector Q_1=(1.4,0,0) persis…
X-ray diffraction experiments under pressure in a diamond anvil cell have been performed to gauge any response of the crystalline lattice of URu2Si2 to the "hidden order" or antiferromagnetic transitions, the latter of which is accessible only with applied pressure. The ambient-pressure crystal structure of URu2Si2 persists to high pressure, and structural characterization reveals a reasonably robust crystal lattice with respect to both of the aforementioned temperature-induced electronic transitions. Coupling between the lattice and the "hidden order" transition is reconfirmed to be subtle, a…
Based on a localized crystal electric field model for the U^{4+} in the (5f)^2-configuration, we analyze the resonant x-ray scattering spectra around U M_{IV} and M_{V} edges in URu_{2}Si_{2}, taking full Coulomb and spin-orbit interactions into account. We consider two level schemes, a singlet model of Santini and Amoretti and a doublet model of Ohkawa and Shimizu, and assume the antiferroquadrupolar order and the antiferromagnetic order as candidates for the ambient pressure phase and the high pressure phase. It is found that the spectral shapes as a function of photon energy are independent…
We have measured the magnetic field dependence of the ordered antiferromagnetic moment and the magnetic excitations in the heavy-fermion superconductor URu2Si2 for fields up to 17 Tesla applied along the tetragonal c axis, using neutron scattering. The decrease of the magnetic intensity of the tiny moment with increasing field does not follow a simple power law, but shows a clear inflection point, indicating that the moment disappears first at the metamagnetic transition at ~40 T. This suggests that the moment m is connected to a hidden order parameter Phi which belongs to the same irreducible…
We develop a Landau Ginzburg theory of the hidden order phase and the local moment antiferromagnetic phase of URu_2Si_2. We unify the two broken symmetries in a common complex order parameter and derive many experimentally relevant consequences such as the topology of the phase diagram in magnetic field and pressure. The theory accounts for the appearance of a moment under application of stress and the thermal expansion anomaly across the phase transitions. It identifies the low energy mode which is seen in the hidden order phase near the conmensurate wavector (0,0, 1) as the pseudo-Goldstone …
We report a comprehensive electronic structure investigation of the paramagnetic (PM), the large moment antiferromagnetic (LMAF), and the hidden order (HO) phases of URu$_2$Si$_2$. We have performed relativistic full-potential calculations on the basis of the density functional theory (DFT), employing different exchange-correlation functionals to treat electron correlations within the open $5f$-shell of uranium. Specifically, we investigate---through a comparison between calculated and low-temperature experimental properties---whether the $5f$ electrons are localized or delocalized in URu$_2$S…
This Topical Review describes the multitude of unconventional behaviors in the hidden order, heavy fermion, antiferromagnetic and superconducting phases of the intermetallic compound URu$_2$Si$_2$ when tuned with pressure, magnetic field, and substitutions for all three elements. Such `perturbations' result in a variety of new phases beyond the mysterious hidden order that are only now being slowly understood through a series of state-of-the-science experimentation, along with an array of novel theoretical approaches. Despite all these efforts spanning more than 30 years, hidden order (HO) rem…
We consider the effects of impurities on the enigmatic hidden order (HO) state of the heavy-fermion material URu2Si2. In particular, we focus on local effects of Rh impurities as a tool to probe the suppression of the HO state. To study local properties we introduce a lattice free energy, where the time invariant HO order parameter "psi" and local antiferromagnetic (AFM) order parameter M are competing orders. Near each Rh atom the HO order parameter is suppressed, creating a hole in which local AFM order emerges as a result of competition. These local holes are created in the fabric of the HO…
We have used specific heat and neutron diffraction measurements on single crystals of URu$_{2-x}$Fe$_x$Si$_2$ for Fe concentrations $x$ $\leq$ 0.7 to establish that chemical substitution of Ru with Fe acts as "chemical pressure" $P_{ch}$ as previously proposed by Kanchanavatee et al. [Phys. Rev. B {\bf 84}, 245122 (2011)] based on bulk measurements on polycrystalline samples. Notably, neutron diffraction reveals a sharp increase of the uranium magnetic moment at $x=0.1$, reminiscent of the behavior at the "hidden order" (HO) to large moment antiferromagnetic (LMAFM) phase transition observed a…
Within the framework of the Kondo-Heisenberg model, we analyse the effect of charge fluctuation on the modulated spin liquid (MSL) and antiferromagnetic (AF) orders which were established in a previous publication. We discuss the emergence of two quantum critical lines separating the coexisting Kondo-MSL and Kondo-MSL-AF phases. The various order parameters of the system possess a characteristic signature observable on the electronic band structure of each of the phases. We calculate that the MSL order is indeed a possible explanation of the hidden order phase of $URu_{2}Si_{2}$ heavy fermion …
We study URu$_{2-x}$Fe$_x$Si$_2$, in which two types of staggered phases compete at low temperature as the iron concentration $x$ is varied: the nonmagnetic "hidden order" (HO) phase below the critical concentration $x_c$, and unconventional antiferromagnetic (AF) phase above $x_c$. By using polarization resolved Raman spectroscopy, we detect a collective mode of pseudovector-like $A_{2g}$ symmetry whose energy continuously evolves with increasing $x$; it monotonically decreases in the HO phase until it vanishes at $x=x_c$, and then reappears with increasing energy in the AF phase. The mode's …
We develop a phenomenological mean field theory describing the hidden order phase in URu$_{\text2}$Si$_{\text2}$ as a nematic of the $\text B_{\text{1g}}$ representation staggered along the $c$-axis. Several experimental features are reproduced by this theory: the topology of the temperature--pressure phase diagram, the response of the elastic modulus $(C_{11}-C_{12})/2$ above the transition at ambient pressure, and orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase. In this scenario, hidden order is characterized by broken rotational symmetry that is modulated along t…
Spontaneous, collective ordering of electronic degrees of freedom leads to second-order phase transitions that are characterized by an order parameter. The notion "hidden order" (HO) has recently been used for a variety of materials where a clear phase transition occurs to a phase without a known order parameter. The prototype example is the heavy-fermion compound URu2Si2 where a mysterious HO transition occurs at 17.5K. For more than twenty years this system has been studied theoretically and experimentally without a firm grasp of the underlying physics. Using state-of-the-art density-functio…
The pressure-temperature phase diagram of the heavy-electron superconductor URu2Si2 has been reinvestigated by ac-susceptibility and elastic neutron-scattering (NS) measurements performed on a small single-crystalline rod (2 mm in diameter, 6 mm in length) in a Cu-Be clamp-type high-pressure cell (P < 1.1 GPa). At ambient pressure, this sample shows the weakest antiferromagnetic (AF) Bragg reflections reported so far, corresponding to the volume-averaged staggered moment of mord ~ 0.011 mB/U. Under applied pressure, the AF scattering intensity exhibits a sharp increase at P ~ 0.7 GPa at low te…
Phase transitions and symmetry are intimately linked. Melting of ice, for example, restores translation invariance. The mysterious hidden order (HO) phase of URu$_2$Si$_2$ has, despite relentless research efforts, kept its symmetry breaking element intangible. Here we present a high-resolution x-ray diffraction study of the URu$_2$Si$_2$ crystal structure as a function of hydrostatic pressure. Below a critical pressure threshold $p_c\approx3$ kbar, no tetragonal lattice symmetry breaking is observed even below the HO transition $T_{HO}=17.5$ K. For $p>p_c$, however, a pressure-induced rotation…
Electrical resistivity measurements were performed on single crystals of URu$_2-x$Os$_x$Si$_2$ up to $x$ = 0.28 under hydrostatic pressure up to $P$ = 2 GPa. As the Os concentration, $x$ , is increased, (1) the lattice expands, creating an effective negative chemical pressure $P_{ch}$($x$), (2) the hidden order (HO) phase is enhanced and the system is driven toward a large-moment antiferromagnetic (LMAFM) phase, and (3) less external pressure $P_{c}$ is required to induce the HO to LMAFM phase transition. We compare the $T(x)$, $T(P)$ phase behavior reported here for the URu$_2-x$Os$_x$Si$_2$ …
Complex electronic matter exhibit subtle forms of self organization which are almost invisible to the available experimental tools, but which have dramatic physical consequences. One prominent example is provided by the actinide based heavy fermion material URu_2Si_2. At high temperature, the U-5f electrons in URu_2Si_2 carry a very large entropy. This entropy is released at 17.5K via a second order phase transition to a state which remains shrouded in mystery, and which was termed a "hidden order" state. Here we develop a first principles theoretical method to analyze the electronic spectrum …
Motivated by the observation of a giant Nernst effect in URu_2Si_2, the thermoelectric response of the related system UPt_2Si_2 was investigated using thermal and electric transport properties such as the Nernst and Seebeck effects, thermal conductivity, Hall effect and electrical resitivity. Unlike URu_2Si_2, UPt_2Si_2 is neither superconducting nor exhibits a ``hidden-order'' state. Nevertheless a pronounced Nernst effect anomaly is found to coincide with the onset of the antiferromagnetic order in UPt_2Si_2. Although the absolute values are substantially lower, its appearance and characteri…
Thermal expansion, electrical resistivity, magnetization, and specific heat measurements were performed on URu$_{2-x}$Fe$_{x}$Si$_{2}$ single crystals for various values of the Fe concentration $x$ in both the hidden order (HO) and large moment antiferromagnetic (LMAFM) regions of the phase diagram. Our results show that the paramagnetic (PM) to HO and LMAFM phase transitions are manifested differently in the thermal expansion coefficient. For Fe concentrations near the boundary between the HO and LMAFM phases at $x_c$ ~ 0.1, we observe two features in the thermal expansion upon cooling, one t…
We address two long-standing questions regarding the hidden order in URu2Si2: Is it associated with the hybridization process, and what are the distinct roles played by the localized and itinerant electrons? Our quasiparticle scattering spectroscopy reveals a hybridization gap ubiquitous in the entire phase space spanned by P and Fe substitutions in URu2Si2, including the no-order and antiferromagnetic regions, with minimal change upon crossing the phase boundary. This indicates its opening isn't associated with the ordering, and thus localized electrons must be the major player. Towards a con…
Novel electronic states resulting from entangled spin and orbital degrees of freedom are hallmarks of strongly correlated f-electron systems. A spectacular example is the so-called 'hidden-order' phase transition in the heavy-electron metal URu2Si2, which is characterized by the huge amount of entropy lost at T_{HO}=17.5K. However, no evidence of magnetic/structural phase transition has been found below T_{HO} so far. The origin of the hidden-order phase transition has been a long-standing mystery in condensed matter physics. Here, based on a first-principles theoretical approach, we examine t…
We carried out simultaneous measurements of electrical resistivity and thermal expansion of the heavy-fermion compound URu2Si2 under pressure using a single crystal. We observed a phase transition anomaly between hidden (HO) and antiferromagnetic (AFM) ordered states at TM in the temperature dependence of both measurements. For the electrical resistivity, the anomaly at TM was very small compared with the distinct hump anomaly at the phase transition temperature T0 between the paramagnetic state (PM) and HO, and exhibited only a slight increase and decrease for the I // a-axis and c-axis, resp…
We have performed neutron scattering experiments on the heavy fermion superconductor URu$_2$Si$_2$ to search for the orbital currents predicted to exist in the ordered phase below {\Tn} = 17.5 K. Elastic scans in the (H, K, 0) and (H, 0, L) planes revealed no such order parameter at low temperatures. This does not completely rule out orbital current formation, because our detection limit for a ring of scattering is 0.06(1) {\ub}, which is greater than the size of the predicted moment of ~0.02 {\ub}. However, on heating, a ring of quasielastic scattering does exist in the (H, K, 0) plane center…
In matter, any spontaneous symmetry breaking induces a phase transition characterized by an order parameter, such as the magnetization vector in ferromagnets, or a macroscopic many-electron wave-function in superconductors. Phase transitions with unknown order parameter are rare but extremely appealing, as they may lead to novel physics. An emblematic, and still unsolved, example is the transition of the heavy fermion compound URu$_2$Si$_2$ (URS) into the so-called hidden-order (HO) phase when the temperature drops below $T_0 = 17.5$K. Here we show that the interaction between the heavy fermio…
We investigate quasiparticle relaxation dynamics in URu$_{2-x}$Fe$_{x}$Si$_{2}$ single crystals using ultrafast optical-pump optical-probe (OPOP) spectroscopy as a function of temperature ($T$) and Fe substitution ($x$), crossing from the hidden order (HO) phase ($x$ = 0) to the large moment antiferromagnet (LMAFM) phase ($x$ = 0.12). At low $T$, the dynamics for $x$ = 0 and $x$ = 0.12 are consistent with the low energy electronic structure of the HO and LMAFM phases that emerge from the high $T$ paramagnetic (PM) phase. In contrast, for $x$ = 0.1, two transitions occur over a narrow $T$ range…
We have performed elastic neutron scattering experiments under uniaxial stress sigma applied along the tetragonal [100], [110] and [001] directions for the heavy electron compound URu2Si2. We found that antiferromagnetic (AF) order with large moment is developed with sigma along the [100] and [110] directions. If the order is assumed to be homogeneous, the staggered ordered moment mu_o continuously increases from 0.02 mu_B (sigma=0) to 0.22 mu_B (0.25 GPa). The rate of increase partial mu_o/partial sigma is ~ 1.0 mu_B/GPa, which is four times larger than that for the hydrostatic pressure (part…
Thermal expansion, magnetostriction and magnetization measurements under magnetic field and hydrostatic pressure were performed on a $\mathrm{UAu_{2}Si_{2}}$ single crystal. They revealed a large anisotropy of magnetoelastic properties manifested by prominent length changes leading to a collapse of the unit-cell volume accompanied by breaking the fourfold symmetry (similar to that in $\mathrm{URu_{2}Si_{2}}$ in the hidden-order state) in the antiferromagnetic state as consequences of strong magnetoelastic coupling. The magnetostriction curves measured at higher temperatures confirm a bulk char…
Novel electronic phenomena frequently form in heavy fermions as a consequence of the mutual nature of localization and itineracy of f electrons. On the magnetically ordered side of the heavy fermion phase diagram, f moments are expected to be localized and decoupled from the Fermi surface. It remains ambiguous whether a Kondo lattice can develop inside the magnetically ordered phase. Using spectroscopic imaging with the scanning tunneling microscope, complemented by neutron scattering, x ray absorption spectroscopy, and dynamical mean field theory, we probe the electronic states in the antifer…
We consider the effective 2+1 dimensional electrodynamics (QED_3) of low-energy quasiparticles coupled to fluctuating vortex loops in the d-wave superconductor, with the velocity anisotropy: v_F not equal to v_Delta. This theory should be relevant to the quantum superconductor-insulator transition in underdoped cuprates. Working in the customary large-N approximation, we find that weak anisotropy is a marginally irrelevant perturbation to the Lorentz invariant QED_3, and that the critical number N_c of Dirac fields below which the theory suffers the antiferromagnetic (chiral) instability stays…
In this paper we introduce an exactly solvable Kondo lattice model without any fine-tuning local gauge symmetry. This model describes itinerant electrons interplaying with a localized magnetic moment via only longitudinal Kondo exchange. Its solvability results from conservation of the localized moment at each site, and is valid for arbitrary lattice geometry and electron filling. A case study on square lattice shows that the ground state is a Néel antiferromagnetic insulator at half-filling. At finite temperature, paramagnetic phases including a Mott insulator and correlated metal are found. …
We report the high-field induced magnetic phase in single crystal of U(Ru0.92Rh0.08)2Si2. Our neutron study combined with high-field magnetization, shows that the magnetic phase above the first metamagnetic transition at Hc1 = 21.6 T has an uncompensated commensurate antiferromagnetic structure with propagation vector Q2 = ( 2/3 0 0) possessing two single-Q domains. U moments of 1.45 (9) muB directed along the c axis are arranged in an up-up-down sequence propagating along the a axis, in agreement with bulk measurements. The U magnetic form factor at high fields is consistent with both the U3+…
We systematically explore the ground-state phase diagram of the Kane-Mele-Kondo lattice model on the honeycomb lattice, in particular, we focus on its magnetic properties which has not been studied in the previous publication[Feng, Dai, Chung, and Si, Phys. Rev. Lett. \textbf{111}, 016402 (2013)]. Beside the Kondo insulator found in that paper, two kinds of antiferromagnetic spin-density-wave phases are identified. One is the normal antiferromagnetic spin-density-wave state and the other is a nontrivial topological antiferromagnetic spin-density-wave state with a quantized spin Hall conductanc…
Vanadium sesquioxide V2O3 is considered a textbook example of Mott-Hubbard physics. In this paper we present an extended optical study of its whole temperature/doping phase diagram as obtained by doping the pure material with M=Cr or Ti atoms (V1-xMx)2O3. We reveal that its thermodynamically stable metallic and insulating phases, although macroscopically equivalent, show very different low-energy electrodynamics. The Cr and Ti doping drastically change both the antiferromagnetic gap and the paramagnetic metallic properties. A slight chromium content induces a mesoscopic electronic phase separa…
The Mott-insulator κ-(BEDT-TTF)_2Cu_2(CN)_3 is the prime candidate of a quantum spin liquid with puzzling magnetic properties. Our THz and infrared investigations reveal that also the charge dynamics does not follow the expectations for a Mott insulator. The frequency-dependent conductivity exhibits a power-law behavior $σ_1(ω)\propto ω^n$ that grows stronger as the temperature decreases and extends all the way through the far-infrared. With $n\approx 0.8$ to 1.5 we obtain a significantly smaller exponent than predicted by Ng and Lee [Phys. Rev. Lett. {\bf 99}, 156402 (2007)]. We suggest fluct…
Remote magnon sensing detects the universal phase stiffness jump in Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$
University of Washington（华盛顿大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of Chemistry and Technology Prague（布拉格化学与技术大学）；Columbia University（哥伦比亚大学）；National Institute for Materials Science（物质材料研究机构）；National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
The algebraic spin liquid is a long-sought-after phase of matter characterized by the absence of quasiparticle excitations, a low-energy description in terms of emergent Dirac fermions and gauge fields interacting according to (2+1)D quantum electrodynamics (QED$_3$), and power-law correlations with universal exponents. The prototypical algebraic spin liquid is the Affleck-Marston $π$-flux phase, originally proposed as a possible ground state of the spin-1/2 Heisenberg model on the 2D square lattice. While the latter model is now known to order antiferromagnetically at zero temperature, recent…
