U. Schollw¨ock; S. Chakravarty (UCLA); J.O. Fjærestad and J.B. Marston (Brown);
M. Troyer (ETH Z¨urich) Strongly correlated quantum systems owe a
large part of their interest to the fact that pure quantum phenomena are less obscured by the averaging nature of thermal fluctuations or ef-fective screening than in many more conven-tional solid state systems. At very low tem-peratures, the possibility to break obvious and hidden symmetries present in those systems promises phase diagrams rich in phases of a rather unique quantum nature. To this day, how-ever, the roster of states with broken symmetries is limited despite the variety of symmetries that can be broken in a strongly correlated electronic system. The difficulty is that it is not always clear what should be the effective Hamiltonian, nor is it clear how a complex quantum order fits into the phase diagram of a real material. Here, we consider an unusual state that spontaneously
breaks time reversal symmetry, in which cur-rents circulating around lattice plaquettes are ar-ranged in a pattern of alternating direction of circulation. As such currents give rise to local magnetic fluxes of alternating direction, such a state is also referred to as a staggered flux state.
In fact, such states have a long history in the physics of correlated electron systems: under the name of orbital antiferromagnets they were first considered in the late sixties in the con-text of excitonic insulators by Halperin and Rice but then discarded in favor of more con-ventional order. After the discovery of the cuprate high-temperature superconductors they were rediscovered by Affleck, Marston, Schulz and Kotliar and baptized staggered flux phases.
Many of their properties were discussed, but
were forgotten again in the absence of experi-mental vindication. The discovery of the pseu-dogap regime has brought them back onto the scene once more: competing with the proposal that the pseudogap is essentially due to inco-herent Cooper pair preformation, attempts have been made by Lee to explain this regime in terms of fluctuations of SF order and a proposal has been made by Chakravarty that it is not fluctuations, but a true broken symmetry that is at the origin of the pseudogap. This ordered state was now called the singlet d-density wave (DDW), because it was recognized that this un-usual state described by spontaneous currents arises naturally as the generalization of density waves to higher angular symmetries. The con-ventional charge-density wave (CDW) in which charge is modulated in space is its s-wave coun-terpart with angular momentum zero. Another type of breakdown of time-reversal symmetry in which the circulating currents obey transla-tional symmetry, as opposed to the staggered DDW, has been pointed out by Varma.
Although much indirect experimental evidence of DDW can be argued to exist, a direct obser-vation of DDW would be Bragg reflection of neutrons carrying magnetic moments from the staggered arrangement, on the scale of a few A, of circulating currents. Recent neutron scat-˚ tering experiments have, however, been contro-versial and more precise and well-characterized experiments are underway to settle this issue.
Thus, a theoretical exploration of microscopic models of correlated electronic systems show-ing such time-reversal symmetry-breakshow-ing with controlled methods has acquired urgency. It is a sobering observation that for the two-dimensional case a rigorous or at least quasi-exact demonstration of the very existence of such a state for some microscopic Hamiltonian is still missing after many years of intense re-search, testifying to the highly complex nature of the problem. To make progress, various au-thors have therefore studied the simplest geo-metrical structure in the form of a two-leg lad-der that can support staggered orbital currents,
Figure 16: Circulating plaquette currents on a ladder, characteristic of the DDW phase.
Previous studies of DDW order in two-leg ladders have used weak-coupling bosoniza-tion/renormalization group (RG) analyses, a density matrix renormalization group (DMRG) analysis of the t-J–model and a half-filled Hubbard-like model, or exact diagonalization of the t-t’-J–model. At half-filling, models with long-range ordered currents have been found both for spinless and spinful fermions. In con-trast, for the relevant case of doped ladders, only exponential decay has been found for all systems considered so far, essentially of the t-J–type with a variety of modifications.
The approach used by us is the essentially exact DMRG method that can be used for arbitrary interaction strength, unlike the weak-coupling bosonization/RG approaches. We break the time reversal symmetry explicitly by applying an infinitesimal source current hjrung1 on one end of the ladder and measure the current induced in the sample. The results of our calcu-lations are striking. Although common t-J–type models do not exhibit long-ranged DDW or-der, a separate class of repulsive Hamiltonians show robust long-ranged DDW order even in the presence of substantial doping. These have their historical origin in a half-filled SO(5) in-variant model on a ladder studied by Lin and Scalapino[Lin et al., Physical Review B 58, 1794 (1998); Scalapino et al., Physical Review B 58, 443 (1998)]. At precise half-filling, it was shown to exhibit DDW in its phase diagram.
We find, however, that SO(5) invariance is ir-relevant by considering coupling constants very far from the ‘SO(5) parameters’ and by substan-tially doping this model. The real reason for success with this class of models is that they al-low a competition between CDW (onsite singlet
states (more precisely rung-singlet states), re-sulting in a local kinetic exchange between them, giving rise to the DDW phase. This is much like as the situation in the cuprates would be in the proposal by Chakravarty et al. in which the DDW phase is an intermedi-ate regime between a multiplicity of complex charge ordered states and DSC.
Our set of low-energy Hamiltonians is defined:
Ht J
UV
t
∑
ijσ
c†i
σcjσhc U 2
∑
i
ni12
∑
r
JSr1Sr2V
nr11nr21 (10)
where r labels rungs and 1 or 2 the leg. In the half-filled case this Hamiltonian has a precise SO(5) symmetry when J= 4 (U + V). But as soon as the system is doped, or the parameters are no longer finely tuned, there is no SO(5)
symmetry. The weak coupling phase diagram at half-filling obtained from bosonization/RG, as shown in Fig. 17, gives us some guidance as to where to look in our DMRG calculations.
Figure 17: The weak coupling phase diagram at half-filling for Ht J
UV
from a bosonization calcula-tion. The open circles correspond to the parameters in Fig. 18.
Figure 18: Rung current jrungras a function of the location of the rung r in a tJ– U – Vmodel at 4% doping on a 1002 ladder, with parameters U = 0.25, t = V= 1 and an edge current of 0.0001. The sequence of figures correspond to (a) J= 0.8, (b) J= 1.1, (c) J= 1.5, and (d) J= 1.7. In (c), we show the profile of the hole density depicted as solid dots corresponding to the scale on the right. Note the vast differences in the scales of the current strengths.
Other than the DDW and CDW, there are two relevant states that can be adiabatically contin-ued to resonating valence bond states of the short-range variety. These states are repre-sented by
The DDW lies between the CDW and the rung-singlet phases.
We have studied the Hamiltonian in Eq.(10) for a range of parameters and find long-range DDW order in the doped model, which has nothing to do with SO(5) symmetry. Nonethe-less, the DDW phase is situated between the CDW and the rung-singlet phases, similar to the weak coupling bosonization results for half-filled ladders. As a typical example, we have shown in Fig. 18 our results for the rung cur-rent as induced by an edge curcur-rent of tiny mag-nitude 0.0001 t. The parameters chosen were U = 0.25, t = V= 1, J= 0.8, 1.1, 1.5, 1.7, and δ= 0.04. As a response, we see robust long-ranged DDW order in the middle of this range
of J with stripe-like features where pairs of holes reside; see, in particular, Fig. 18(c), where we also plot the hole density, and the coex-istence with stripe order is especially evident from the antiphase domain wall structure. The induced currents clearly alternate in sign and can be of order unity, in units of t, even though the source current is infinitesimally small. For ladders of lengths 100, 150, and 200, and for parameters of Fig. 18(c), the current amplitudes are 0.56, 0.53, and 0.53 respectively, consis-tent with long-range order, though in a numeri-cal numeri-calculation it is never possible to rule out a very slow decay. We have studied the d-wave pairing correlations, and find only exponential decay in ladders that exhibit DDW long-range order. We also find a robust spin gap. For suf-ficiently strong doping, roughly between 10 to 20%, DDW is suppressed.
In summary, we have shown for the first time in a quasi exact fashion that there are repul-sive microscopic models that exhibit DDW or-der at finite doping, providing an insight into the physical origin of this symmetry-breaking.
The challenge is now, of course, to bridge the gap between effective Hamiltonians as consid-ered above, and real substances.