Statistical transmutation in doped quantum dimer models.
We prove a “statistical transmutation” symmetry of doped quantum dimer models on the square, triangular and kagome lattices: the energy spectrum is invariant under a simultaneous change of statistics (i.e. bosonic into fermionic or vice-versa) of the holes and of the signs of all the dimer resonance loops. This exact transformation enables to define duality equivalence between doped quantum dimer Hamiltonians, and provides the analytic framework to analyze dynamical statistical transmutations. We investigate numerically the doping of the triangular quantum dimer model, with special focus on the topological dimer liquid. Doping leads to four (instead of two for the square lattice) inequivalent families of Hamiltonians. Competition between phase separation, superfluidity, supersolidity and fermionic phases is investigated in the four families.
The discovery of exotic liquids such as topological spin liquids (SL) Wen (); Moessner_prl_2001_Z2 () is one of the challenges of modern condensed matter physics. Anderson proposed that the parent (insulating) state of the high-temperature superconductors is in fact a SL, the resonating valence bond (RVB) state and that spin-charge separation (and superconductivity) will occur under doping AndersonRVB (): the original electron fractionalizes into two emergent particles, a holon carrying the charge quantum and a spinon carrying the spin quantum. Although the original electron is a fermion, there has been a long-standing debate regarding the actual statistics of holons and spinons in such a “deconfinement” scenario. In this context, Rokhsar and Kivelson RK () introduced the quantum dimer model (QDM) as a simple effective model to describe magnetically disordered phases. The basis assumption here is that the spinon spectrum is gapped Read+sachedv_1991 () (strictly speaking, in the QDM it is infinite) and the dimers between nearest neighbor (NN) sites mimic fluctuating SU(2) singlets of paired electrons. The underlying microscopic exchange interaction leads to effective attraction/repulsion between dimers and dimer-flips along closed loops (see below). Doped QDM, where dimers (i.e. pairs of electrons) are removed from the system, leading to itinerant holes, have also been studied RK (); Kivelson_1989 (); RMP0 (); poilblanc_2008 (). There, holes and dimers are strongly coupled due to hard-core constraints. Variants of doped QDM have been also constructed to physically describe polarized spinons induced by a magnetic field RMP (). Naively, one expects holons to be of fermionic nature (while spinons should be bosonic) but, in spite of this naive expectation, the statistics of holons has been debated over the last 20 years. Earlier work suggested that holon excitations were fermions Read_1989 (). It was then argued that, in fact, holon statistics is dictated by energetic considerations and under some conditions a holon could become a bosonic composite through binding of a flux quantum Kivelson_1989 (), the “vison” in the QDM language. This was indeed observed recently by exact diagonalization techniques in the square lattice poilblanc_2008 () where, by varying the ratio of dimer kinetic energy vs holon kinetic energy, one can transit from a regime where low-energy quasi-particles behave as fermions to a regime where they behave as bosons.
In this letter we prove an exact duality transformation that provides a framework to analyze this dynamical statistical transmutation. More precisely, we study the generic Hamiltonians of QDM’s doped with holes depicted in Fig. 1, which contain three contributions, and respectively the flipping and potential energy terms for the dimers of amplitudes and and the hopping term for vacancies (named “holes”) of amplitude .
The Hilbert space corresponds to all dimer coverings with a fixed hole density. The potential term is proportional to the number of flippable plaquettes (which is not conserved by dimer flips and hole motion). Special attention is devoted to frustrated lattices, the triangular lattice with edge-sharing triangular units and the kagome lattice with corner-sharing triangular units, which we compare to the results obtained for the square lattice poilblanc_2008 (). The exact symmetry of these Hamiltonians states that their spectrum is invariant under simultaneous transformations of the statistics of the holes, bosons into fermions or vice-versa, , and of the sign of (all) dimer kinetic amplitudes, , and being unchanged.
Equivalence classes – For fixed values of the magnitudes of , and , one can define eight different Hamiltonians by changing the signs of and/or and by choosing bosonic or fermionic statistics for the mobile holes. Using the exact symmetry mentioned above and proven below, these eight different Hamiltonians can be grouped in pairs with identical spectra, hence defining four non-equivalent families. This is summarized in Fig. 2.
More precisely, the established equivalence is between and , both represented by the family label (a) and in the same way for the pairs of Hamiltonians and with corresponding respectively to families (b), (c) and (d). We argue that such an equivalence is a general feature of doped-two-dimensional QDMs, independently of the lattice geometry. Physically, classes (a) and (b) ((c) and (d)) describe systems with doped polarized spinons (doped holons), and are called “spinonic” QDMs (“electronic” QDMs). Note that, for the square (or hexagonal) lattice, an additional symmetry reduces the number of non-equivalent families from 4 to 2. In Fig. 2, the two families defined for the square lattice by on one hand, and by on the other hand, were dubbed as “Perron-Frobenius” and “non-Perron-Frobenius” Hamiltonians and studied in poilblanc_2008 () while, for the triangular lattice, only the unfrustrated Hamiltonian (representing family (a)) was studied in RMP ().
As the transformation we shall establish below change bosons into fermions (and vice versa), we can start by assuming, without lost of generality, that the (bare) holons in the doped QDM are bosons. We implement a two dimensional Jordan-Wigner transformation on these bosons to change their statistics to fermionic. In contrast to one dimensional systems, this resulting transformed Hamiltonian is highly non-local Fradkin_prl_1989 (); Wang_1991 (); Cabra_JW (); Lamas_2006 () requiring, in general, a mean field approximation to proceed further, at least from an analytical point of view. Next, we show that the non-local terms can be absorbed by using a different representation for the dimer operators, which keeps their bosonic character. This key feature provides then an elegant proof of the “statistical transmutation symmetry” of these models.
Proof of statistical transmutation symmetry – It is convenient to write the Hamiltonian in a second quantized form, by introducing creation operators for a dimer siting between sites and and holes by operators . In our conventions, dimers between sites and are created by spatially symmetric operators , both operators ( and ) are bosonic and mutually commuting. It is instructive to notice that the dimer bosonic operators can be thought as bilinears of “electrons” operators: . In terms of these operators we implement the hard-core constraint where the sum runs over NN of site . Let us call the projector on the subspace where the constraint holds. In the following we use systematically the projected Hamiltonian ( since all terms of commute with ), which prove to be very useful later.
Let us now apply 2-D Jordan-Wigner (JW) transformation on the holon operators Wang_1991 ():
where and is the complex coordinate of the j-th hole. Using that it follows immediately that two -operators in different sites anticommute and the constraint and the phase can be written equally in terms of or operators.
In order to understand the consequences of transformation (1) on the Hamiltonian, the hopping of holons can be written, for an arbitrary lattice, as a sum of three-site Hamiltonians
Making use of the transformation (1) we obtain
In the last equation we have changed boson operators by fermionic ones at the cost of introducing non-local interactions. In other words we have written a boson as a composite particle consisting of an electron with an attached flux. As we show just below, this non-local interaction can be absorbed in a redefinition of the dimer operators without affecting their bosonic commutations relation. In order to define new dimer operators including the phases in their definition, we must be able to write in terms of operators . This can be performed in the following way: When applied into the subspace projected by , we can change by in the exponentials, where
The remaining non-local exponential operators are re-written in terms of which can be absorbed by defining . As announced above, it is a simple matter to see that operators are also bosonic, as it should be to make sense as dimer operators. Then, transformation (1) together with the definition of allow us to change the statistics of holes from bosonic to fermionic. After this transformation the hopping term is written in terms of operators and and the hopping amplitude changes to . Let us now investigate the effect of this transformation specifically for each lattice :
i) Square lattice. – The Hamitonian is defined by , where:
Before going to the details of the Jordan-Wigner transformed Hamiltonian let us mention some well known gauge transformations which can be performed. For zero doping, a simple gauge transformation on the dimers can be done to show the equivalence of the Hamiltonians with and . In the case of non-zero doping, changing the sign of the holes wave function with a wave vector transformed as one obtains easily the equivalence between Hamiltonians with and . We can now proceed to the computation of the Jordan-Wigner transformed Hamiltonian. The transformed Hamiltonian becomes, after some algebra SM (), , where the tilde means that dimers and holes are created by operators and respectively. In the dimer kinetic Hamiltonian the amplitude has been changed by .
ii) Triangular lattice. – Here the Hamiltonian is written using:
and similar expressions for and for the other orientations of the rhombi Albuquerque (). corresponds to the holon hopping on the up-triangles, the total hopping Hamiltonian being . It is important to note that, on such non-bipartite lattices, the equivalence does not hold anymore, as verified numerically. This implies that there are now four families of equivalent Hamiltonians, instead of two for the square case. On the triangular lattice, the dual Hamiltonian reads , with the same convention as before SM (). A corollary of this result is that, as for the square lattice, the Hamiltonians with and are equivalent in the zero doping case.
iii) Kagome lattice. – On this geometry, all dimer resonant loops (of length ) involving a single hexagon become important and should be considered PMS (). Nearest-neighbor hole hopping as depicted in Fig. 1 can be introduced as in Ralko_Poilblanc_kagome () (the reader can refer to PMS (); Ralko_Poilblanc_kagome () for more details). The JW transformed Hamiltonian is obtained in the same way as before and since it is interesting on its own, it will be presented elsewhere large (). Here, we only summarize the main results. By using the JW transformation followed by (more involved) gauge transformations on dimer and hole operators, note:gauge_transformation () one can write the dual Hamiltonian with all kinetic dimer amplitudes changed into . As before, dimers and holons are created by and respectively.
Statistical transmutation and choice of Hamiltonian – The above statistical transmutation symmetry is of great importance in analytic and numerical investigations of doped QDMs, establishing their phase diagrams. In poilblanc_2008 (), it was shown that, in the square lattice, by varying the ratio or the doping, the system undergoes a series of phase transitions. One of those phase transitions corresponds to a dynamical transmutation, between a phase where elementary low-energy quasi-particles are fermionic holes, to a phase where the low-energy quasi-particles acquire a bosonic nature.
It was in fact predicted that holon and vison could pair up, leading to such statistical transmutation Kivelson_1989 (). Then, if the microscopic Hamiltonain is chosen such that holons are fermions, the fermionic phase corresponds to a weak-coupling regime and the bosonic phase to a strong-coupling one. This picture gets interchanged if holons in the microscopic Hamiltonian are chosen to be bosons. Hence, using this duality equivalence one can always choose the most relevant microscopic QDM Hamiltonian to be in a weak-coupling regime (depending on the point in the phase diagram).
Phase diagrams – We now complement the exact results with a numerical study of the phase diagram of the four non-equivalent families of Hamiltonians in the triangular lattice where a topological () liquid can be stabilized at zero doping Moessner_prl_2001_Z2 (); Ralko_2005 (). The statistics of the dressed excitations is studied using the method developed in poilblanc_2008 () by investigating the node content of the wave functions. Family (a) in Fig. 3 is the only unfrustrated case and was studied with Green Function Monte Carlo methods in RMP (). In this model bare holons are bosonic and remain representative of the physical excitations in the entire region of the phase diagram, as happen also in the Perron-Frobenius square lattice version, studied in poilblanc_2008 (). The unfrustrated model shows a superfluid phase in a wide range of the phase diagram.
The situation is even more interesting when is changed into , or equivalently, bosons are changed into fermions (families (c) and (d) in Fig. 3). Most of the phase diagram is occupied by what we dubbed a “complex phase” as the statistics of dressed excitations does not correspond solely to bosons or fermions. Similarly to the square lattice, we find a phase separated region at low doping. Family (d) also shows, in addition, an interesting fermion reconstruction at large doping, which we call “fermionic phase”.
In order to provide a better understanding of the conducting phases of the triangular lattice, we insert in the torus an Aharonov-Bohm flux of strength with (achieved by adding a phase shift in e.g. the x-direction) and where the elementary magnetic flux. As done in Ref. RMP0 () on the square lattice, one can reduce the finite-size effects by considering arbitrary boundary conditions in the -direction. A superfluid is characterized by well-defined minima in the ground state energy separated by a finite barrier in the thermodynamic limit. A contrario, a typical signature of (weakly interacting) fermions, a flat energy profile is expected even on such a small cluster Poilblanc1991 (). Here, we report that the ground state energy has well-defined minima quantized at half a flux quantum for all family of models at , compatible with a bare charge superconductor 2e (); RMP0 ()(see inset of Fig. 3(c)). Further detailed discussions on the charge of the “low energy” particles in the superconducting phase will be provided elsewhere large ().
Discussion and perspectives – In this paper we have shown that the nature of the excitations in doped QDM is a much more subtle question than what one would naively think. We have rigorously established equivalence classes between QDM Hamiltonians (see Fig. 2). We claim that this duality relation is a generic feature of QDM independently of the details of the lattice.
This proven duality relation provides a powerful tool to identify the nature of the dressed hole excitations. Indeed, imagine one considers a specific system, with given microscopic parameters, corresponding to one of the four (or two) families of Hamiltonians. A priori two (or four) representations of this system can be chosen freely using the equivalence relations of Fig. 2. As we have shown, there is in fact a specific representation which corresponds to a weakly coupled regime, i.e. where the chosen statistics of the bare holes is the one of the true dressed excitations. This is the representation to be chosen to study e.g. the effect of further perturbations of the system.
Lastly, we have shown numerically that models on the triangular lattice corresponding physically to spinon or holon doping have very different phase diagrams with either a superfluid or a “complex” phase (in which dressed excitations can not be understood solely in terms of fermionic or bosonic degrees of freedom), respectively. In the latter case, we provide evidence of statistical transmutation, analogously to the case of the square lattice.
Acknowledges — AR, DP and PP acknowledge support by the “Agence Nationale de la Recherche” under grant No. ANR 2010 BLANC 0406-0. CAL and DCC are partially supported by CONICET (PIP 1691) and ANPCyT (PICT 1426).
- (1) X.-G.Wen, Adv. in Phys. 44, 405 (1995) and references therein.
- (2) R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001)
- (3) P. W. Anderson, Science 235, 1196 (1987); P.W. Anderson, Mater. Res. Bull. 8, 153 (1973); P. Fazekas and P.W. Anderson, Philos. Mag. 30, 23 (1974).
- (4) D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988).
- (5) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991).
- (6) A. Ralko, F. Mila and D. Poilblanc, Phys. Rev. Lett. 99, 127202 (2007); see also D. Poilblanc et al., Phys.Rev. B 74, 014437(2006).
- (7) D. Poilblanc, Phys. Rev. Lett. 100, 157206 (2008).
- (8) S. Kivelson, Phys. Rev. B 39, 259 (1989).
- (9) A. Ralko, F. Becca and D. Poilblanc, Phys. Rev. Lett. 101, 117204 (2008);
- (10) N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).
- (11) Eduardo Fradkin, Phys. Rev. Lett. 63, 322 (1989).
- (12) Y.R. Wang, Phys. Rev. B 43, 3786 (1991).
- (13) See Supplemental Material at [URL to be inserted by publisher] for the technical details.
- (14) D.C. Cabra, G.L. Rossini. Phys. Rev. B 69, 184425 (2004).
- (15) C.A. Lamas, D.C. Cabra, M.D. Grynberg, G.L. Rossini. Phys. Rev. B 74, 224435 (2006).
- (16) A.F. Albuquerque, H.G. Katzgraber, M. Troyer, G. Blatter, Phys. Rev. B 78, 014503 (2008)
- (17) D. Poilblanc, M. Mambrini and D. Schwandt, Phys. Rev. B 81, 180402(R) (2010).
- (18) D. Poilblanc and A. Ralko, Phys. Rev. B 82, 174424 (2010).
- (19) C.A. Lamas, A. Ralko, D.C. Cabra, D. Poilblanc, M. Oshikawa and P. Pujol, (unpublished)
- (20) The gauge transformation allows to bring the hopping term back to its original form after changes in the hopping amplitudes generated by the statistical transmutation. For convenience, details will be published in large ().
- (21) A. Ralko, M. Ferrero, F. Becca, D. Ivanov and F. Mila, Phys. Rev. B 71, 224109 (2005).
- (22) S. A. Kivelson, D. S. Rokhsar and J. P. Sethna, Euro. Phys. Lett. 6, 353 (1988).
- (23) D. Poilblanc, Phys. Rev. B 44, 9562 (1991).