Microscopic spinon-chargon theory of magnetic polarons in the t-J model
MMicroscopic spinon-chargon theory of magnetic polarons in the t − J model Fabian Grusdt,
1, 2, ∗ Annabelle Bohrdt,
2, 1 and Eugene Demler Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Department of Physics and Institute for Advanced Study,Technical University of Munich, 85748 Garching, Germany (Dated: January 7, 2019)The interplay of spin and charge degrees of freedom, introduced by doping mobile holes intoa Mott insulator with strong anti-ferromagnetic (AFM) correlations, is at the heart of stronglycorrelated matter such as high- T c cuprate superconductors. Here we capture this interplay in thestrong coupling regime and propose a trial wavefunction of mobile holes in an AFM. Our methodprovides a microscopic justification for a class of theories which describe doped holes moving in anAFM environment as meson-like bound states of spinons and chargons. We discuss a model of suchbound states from the perspective of geometric strings, which describe a fluctuating lattice geometryintroduced by the fast motion of the chargon. This is demonstrated to give rise to short-rangehidden string order, signatures of which have recently been revealed by ultracold atom experiments.We present evidence for the existence of such short-range hidden string correlations also at zerotemperature by performing numerical DMRG simulations. To test our microscopic approach, wecalculate the ground state energy and dispersion relation of a hole in an AFM, as well as the magneticpolaron radius, and obtain good quantitative agreement with advanced numerical simulations atstrong couplings. We discuss extensions of our analysis to systems without long range AFM orderto systems with short-range magnetic correlations. I. INTRODUCTION AND OVERVIEW
Despite many years of intense research, key aspects ofthe phase diagram of the Fermi-Hubbard model remainpoorly understood [1, 2]. Simplifying variational theo-ries are lacking in important parameter regimes, whichmakes the search for a unified field theory even morechallenging. In other strongly correlated systems, mi-croscopic approaches have proven indispensable in thedevelopment of field-theoretic descriptions. A prominentexample is constituted by the composite-fermion theoryof the fractional quantum Hall effect, which provides anon-perturbative but conceptually elegant explanation ofa class of topologically ordered ground states [3–5].In the case of the 2D Fermi-Hubbard model at strongcouplings, already the description of a single hole dopedinto a surrounding AFM represents a considerable chal-lenge. This problem is at the heart of high-temperaturesuperconductivity and strongly correlated quantum mat-ter, where pronounced AFM correlations remain presentat short distances even for relatively large doping. Whilemany properties of a single hole moving in an AFM spinbackground are known, their derivation, in particular atstrong couplings, requires sophisticated numerical simu-lations [6–18]. This includes such basic characteristics asthe shape of the single-hole dispersion relation.One of the central obstacles in the search for a gen-erally accepted theory of strongly correlated materials,and the rich phase diagram of high-temperature super-conductors in particular, is the lack of a known unify-ing physical principle. Arguably one of the most influ-ential approaches is the resonating-valence bond (RVB) ∗ Corresponding author email: [email protected] paradigm suggested by Anderson [19]. While it yieldssatisfactory results at intermediate and large doping lev-els [20], it is not powerful enough to accurately describethe low-doping regime, starting on the single-hole level,or capture the disappearance of antiferromagnetism ob-served upon doping.Recent experiments with ultracold fermions in opticallattices [21–24] suggest a new paradigm: By introducingshort-ranged hidden string order, a connection has beendemonstrated between the Fermi-Hubbard model at fi-nite doping and an AFM parent state at half filling, bothin 1D [25, 26] and recently also in 2D [27, 28]. While the1D case can be rigorously proven [29–31], it was arguedthat this hidden string order also emerges in 2D as animmediate consequence of the hole motion [32–38].In this article, we demonstrate that the hidden stringorder paradigm discussed in Ref. [28] provides a unifiedunderstanding of the properties of a single hole dopedinto an AFM parent state, with or without long-rangemagnetic order. We use numerical density-matrix renor-malization group (DMRG) simulations on 8 × t − J model, see Fig. 1. This observationleads us to a microscopic description of individual holesdoped into general spin backgrounds with strong AFMcorrelations.Specifically, we propose a trial wavefunction for themagnetic polaron formed by a single hole, which goesbeyond the RVB paradigm by explicitly including short-range hidden string order. While most theories start fromthe weak coupling regime, where the tunneling rate t ofthe hole is small compared to the super-exchange energy J , our method works best at strong couplings, where a r X i v : . [ c ond - m a t . s t r- e l ] J a n FIG. 1.
Meson-like spinon-chargon bound states andshort-range hidden string order.
A single hole in a 2DAFM forms a meson-like bound state of a spinon and a char-gon, similar to quark-antiquark pairs forming mesons in highenergy physics. (a) In the t − J model, a spinon can be boundto a chargon by a geometric string of displaced spins. (b) Sig-natures of such strings (Σ) can be visualized in individualFock configurations by analyzing the difference to a classi-cal N´eel pattern. We use the matrix product state formalism(MPS) and the DMRG algorithm to generate snapshots ofthe T = 0 ground state of the t − J model with a single hole,similar to the recent measurements using ultracold fermions[28]. In (c) we show the distribution function of the length (cid:96) of string-like patterns emanating from the hole. A strikingdifference is observed between a localized and a mobile hole(MPS, indicated by symbols connected with dashed lines).Mobile holes are described quantitatively by the geometricstring theory (FSA, shaded ribbons) which is based on thestring length distributions p FSA (cid:96) shown in the inset. the bandwidth of a free fermion W t = 8 t is much largerthan the energy range covered by the (para-) magnonspectrum W J ≈ J , i.e. t (cid:29) J/
4. This coincides withthe most relevant regime in high-temperature cuprate su-perconductors, where t ≈ J [1]. Note that we require J/t ≥ .
05 however, below which the Nagaoka polaronwith a ferromagnetic dressing cloud is realized [18, 39].Extensions of our approach to weak couplings, t (cid:46) J , arepossible and will be devoted to future work. A centralpart of our study is the analysis of string patterns in in-dividual images. This contains more information thanthe commonly used two point correlation functions andis motivated by recent experiments with quantum gasmicroscopes.This paper is organized as follows. In the remainder ofthe introduction, we provide a brief review of the knownproperties of magnetic polarons along with an overview ofour new results, concerning in particular the short-range hidden string order and the magnetic polaron radius. InSec. II we discuss our microscopic model for describingindividual dopants in an AFM and introduce the trialwavefunction. In Sec. III we present our numerical resultsand analyze the accuracy of the trial wavefunction. Weclose with an outlook and a discussion in Sec. IV. A. Magnetic polarons
When a single dopant is introduced into a spin back-ground, it can be considered as a mobile impurity whichbecomes dressed by magnetic fluctuations and forms anew quasiparticle – a magnetic polaron. In the caseof a doublon or a hole doped into a Heisenberg AFM,commonly described by the t − J model, the dressingby magnon fluctuations leads to strongly renormalizedquasiparticle properties [6–8]. Here we provide a briefreview of these known properties and their most com-mon interpretation:(i) The dispersion relation of the hole is strongly renor-malized, with a bandwidth W ∝ J rather than thebare hole hopping t ;(ii) The shape of the dispersion differs drastically fromthat of a free hole, − t [cos k x + cos k y ]. It has aminimum at k = ( π/ , π/
2) and disperses weaklyon the edge of the magnetic Brillouin zone (MBZ), | k x | + | k y | = π ; see Fig. 5;(iii) At strong couplings the ground state energy de-pends linearly on J / t / and approaches − √ t when J → t . Theproperties in (iii) can be obtained from numerical cal-culations within the magnetic polaron theory, but theirrelation to an underlying physical mechanism is not madeexplicit. B. Parton picture: Spinons, chargons and strings
As reviewed next, the established properties of mag-netic polarons (i) - (iii) follow more naturally from aspinon-chargon ansatz. Here the magnetic polaron isunderstood as a composite bound state of two partons:A heavy spinon carrying the spin quantum number ofthe magnetic polaron, and a light chargon carrying itscharge. This parton picture of magnetic polarons wasfirst suggested by B´eran et al. [40]. Based on an evenbroader analysis these authors conjectured that mag-netic polarons are composites, closely resembling pairsof quarks forming mesons in high-energy physics.
FIG. 2.
Dressing cloud of magnetic polarons.
Using theMPS formalism and DMRG we calculate local spin correlationfunctions C zn ( d ) = (cid:104) ˆ n h r h ˆ S z r ˆ S z r (cid:105) / (cid:104) ˆ n h r h (cid:105) , where n = 1 ( n = 2)denotes nearest neighbor (diagonal next-nearest neighbor)configurations of r , r , as a function of the bond-center dis-tance d = | ( r + r ) / − r h | of the two spins from the hole. Atstrong coupling the distortion of the magnetic spin environ-ment around the hole follows an almost universal shape whichis described well by the geometric string approach (FSA), ex-cept for some additional features between d ≈ . d ≈ . S z tot = 1 / The properties (i) and (ii) can be understood fromthe parton ansatz by noting that the chargon fluctuatesstrongly on a time scale ∼ /t , whereas the center-of-mass motion of the spinon-chargon bound state is deter-mined by the slower time scale ∼ /J of the spinon [40].Hence the overall kinetics of the bound state is dominatedby the spinon dispersion, which features (i) a band-width W ∝ J ; and (ii) a near-degeneracy at the magnetic zoneboundary because the spinon dynamics is driven by spin-exchange interactions on the bi-partite square lattice.As also previously recognized [32–37], the third result(iii) is related to the string picture of holes moving in aclassical N´eel state [41]: The hole motion creates stringsof overturned spins, leading to a confining force [32].The approximately linear string tension ∝ J leads to thepower-law dependence ∝ J / t / of the ground state en-ergy, and the asymptotic value − √ t is a consequenceof the fractal structure of the Hilbert space defined bystring states [32, 33, 38].It is generally acknowledged that strings play a rolefor the overall energy of magnetic polarons in the t − J model (iii), but the string picture alone does not accountfor the strongly renormalized dispersion relation of thehole, i.e. properties (i) and (ii) above. Hence a completedescription of magnetic polarons needs to combine theparton ansatz with the string picture. As also noted by B´eran et al. [40], it is natural to assume that the stringsare responsible for binding spinons to chargons.While the combination of spinons, chargons and stringsprovides a satisfactory picture of magnetic polarons, aquantitative microscopic description of the meson-likebound states and the underlying partons has not beenprovided. More recently, toy models have been discussedwhich also contain spinon-chargon bound states from thestart [42] and capture the most important physical fea-tures of the pseudogap phase, see also Ref. [43]. But inthese cases, too, the precise connection to the microscopic t − J model and the spinon-chargon binding mechanismremain subject of debate.In this article we introduce a complete microscopicdescription of magnetic polarons, in terms of individ-ual spinons, chargons and so-called geometric strings[28, 38, 44] of displaced – rather than overturned – spinsconnecting the partons. This leads us to a new trialwavefunction of magnetic polarons, which can be con-structed for arbitrary doped quantum AFMs, with orwithout long-range order. Our microscopic descriptionimplies new experimental signatures, which will be dis-cussed next. They go beyond the capabilities of tradi-tional solid state experiments, but can be accessed withultracold atoms.This article extends our earlier work on the spinon-chargon theory for the simplified t − J z model with Isingcouplings between the spins, where the spinon motion isintroduced by Trugman loops [34, 38], and in systemswith mixed dimensionality where the hole motion is con-strained to one direction [44]. Instead of invoking gaugefields for modeling the strings connecting spinons andchargons [40], our approach has a purely geometric ori-gin and generalizes the concept of squeezed space usedto describe doped 1D systems [25, 29–31]. C. Geometric paradigm: Short-range hidden stringorder in magnetic polarons
The properties (i) and (ii) of magnetic polarons dis-cussed in Sec. I A have been measured in solid state ex-periments [45] by spectroscopic techniques. Quantum gasmicroscopy allows to go beyond such measurements andanalyze individual experimental snapshots, obtained inquantum projective measurements, and directly searchfor signatures of string formation [32–37] in real space[38]. This has been done in Ref. [28], where string pat-terns were analyzed and signatures for hidden AFM cor-relations have been obtained.To identify string patterns, the experimental images inRef. [28] were compared to a perfect checkerboard config-uration as expected for a classical N´eel state. String likeobjects were revealed from difference images where thesites deviating from the perfect N´eel pattern are identi-fied, see Fig. 1 (b). Then the distribution of lengths ofsuch string-like objects was analyzed in Ref. [28]. Thestring patterns exist even at half filling due to fluctua-tions of the local staggered magnetic moment, providinga background signal. Upon doping, a significant increaseof the number of string patterns was detected which isproportional to the number of holes [28].Here we perform DMRG simulations on a 8 × t − J model with exactly one hole, | Ψ (cid:105) . To generate snap-shots {| α n (cid:105)} of the wavefunction, we employ MetropolisMonte Carlo sampling of Fock basis states | α (cid:105) with onehole and calculate the required overlap |(cid:104) α | Ψ (cid:105)| usingthe matrix product state formalism. For every generatedsnapshot | α n (cid:105) we calculate the difference to a classicalN´eel pattern and determine the length (cid:96) of non-branchingstring-like defects emanating from the hole, see Fig. 1(b). Cases where no strings exist count as (cid:96) = 0, andin cases with multiple strings the longest object is con-sidered. Our analysis is similar to the experimental onein Ref. [28], except for the simultaneous spin and chargeresolution which leads to a reduction of the backgroundsignal from undoped regions.In Fig. 1 (c) we show our results for the full countingstatistics of string lengths (cid:96) . The DMRG calculationswere performed on a 8 × S z tot = 1 /
2. We compare the case of a mobilehole at weak, t = 0 . J , and strong coupling, t = 3 J , toa Heisenberg AFM with one spin removed, correspond-ing to a localized hole. In the latter case, the majorityof strings has length (cid:96) = 0. Other string lengths arealso found, but even string lengths (cid:96) = 2 , , ... are morelikely than odd ones. This is understood by noting thatquantum fluctuations on top of the classical N´eel stateare caused by spin-exchange terms: individually, theychange the string length by two units.We observe that the string length distributions ob-tained for mobile holes are significantly broader, and theeven-odd effect explained by quantum fluctuations of thespins is much less pronounced. Already for t ≈ J we findthat approximately half of the observed string patternshave a length (cid:96) ≥
2. For t = 3 J the string length distribu-tion continues to broaden and develops a local maximumat (cid:96) = 1. In Fig. 3 (a) we plot the average string length,i.e. the first moment of the distribution. As expectedfrom the linear string tension [32], it depends linearly on( t/J ) / in the strong coupling regime.The string patterns revealed in Fig. 1 (c) in the groundstate share the same characteristics as the string patternsfound experimentally in Ref. [28] over a wide range ofdopings. As in the experiment, we will show in this articlethat the string patterns are described quantitatively bythe so-called frozen spin approximation (FSA), which willbe discussed in detail in Sec. II A. The main assumptionof the FSA is to consider only charge fluctuations alongstrings of displaced spins. While the quantum state ofthe surrounding spins remains unmodified, the positionsof the spins in the lattice change.This geometric paradigm is at the heart of the FSA FIG. 3.
Magnetic polaron radius.
We calculate the size ofthe magnetic polaron as a function of t/J : (a) by determiningthe average length (cid:104) (cid:96) (cid:105) of the string-like objects revealed inindividual snapshots, and (b) by fitting the dependence oflocal spin correlations C n ( d ) on the bond center distance bya function of the form C ∞ n + ae − d/R mp and interpreting thefit parameter R mp as the magnetic polaron radius. For smallvalues of t , the fit to C z ( d ) is not meaningful and we do notprovide any data points in this regime. and allows to construct a new set of snapshots {| α FSA n (cid:105)} describing a mobile hole, starting from a set of snap-shots {| α n (cid:105)} for the undoped system: From the latterwe generate the FSA snapshots by removing a spin inthe center of the cylinder and moving the resulting holein random directions l times, where l is sampled fromthe string length distribution p FSA l obtained from micro-scopic considerations, see Sec. II A. This motion of thehole displaces the spins and introduces short-range hid-den string order.In Fig. 1 (c) we analyze the string patterns in the FSAsnapshots and find remarkable agreement with our fullnumerical DMRG simulations of the mobile holes. Thisremains true for a wide range of parameters t/J . Similaragreement was reported in ED simulations of a simplified t − J model with mixed dimensionality [44]. The inset ofFig. 1 (c) shows the underlying FSA string-length distri-butions p FSA l , which share the same qualitative featuresas the detected string patterns in the main panel. D. Dressing cloud of magnetic polarons
The capability of ultracold atom experiments to resolvethe collapsed quantum state with full resolution of spinand charge simultaneously [21] has recently lead to thefirst microscopic observation of the dressing cloud of amagnetic polaron [47]. The measurements are consistentwith earlier theoretical calculations at zero temperature[9] and show that the local spin correlations are onlyaffected in a relatively small radius of one to two latticesites around the mobile dopant. In Fig. 2 we performsimilar calculations using DMRG, see also Ref. [18], andobserve that the spin correlations approach a universalshape which becomes nearly independent of t/J at strongcouplings, t (cid:29) J/ t/J at finite temperature [26, 47]. Such behav-ior can be understood from the FSA by noting that thecharge is located at one end of the fluctuating geometricstring, which interchanges the sub-lattice indices of thesurrounding spins and hides the underlying AFM corre-lations. For C , C denoting nearest and next-nearestneighbor spin correlations in the undoped system, theFSA predicts diagonal next-nearest neighbor correlators C ( d = 1 / √
2) directly next to the dopant given by C (1 / √ | FSA ≈ (cid:18) p FSA0 + 1 − p FSA0 (cid:19) C + 1 − p FSA0 C . (1)Here d = 1 / √ p FSA l is the string-length distri-bution derived from the FSA approach in Sec. II A, whichis shown in the inset of Fig. 1 (c).The correlations between the mobile hole and the sur-rounding spins are liquid like, with no significant effecton lattice sites more than two sites away even when t/J is large: If we fit an exponential C ∞ n + ae − d/R mp to C n ( d ),we find that R mp – which we identify as the polaron ra-dius – depends only weakly on t/J and the index n of thecorrelator, see Fig. 3 (b). The average string length (cid:104) (cid:96) (cid:105) of the string patterns revealed in individual microscopicFock configurations, in contrast, depends more stronglyon t/J , see Fig. 3 (a).In the string picture, these liquid like correlations inthe local spin environment of the mobile dopant are adirect consequence of the large number of string config-urations N Σ ( (cid:96) ) with a specific string length (cid:96) , growingexponentially: N Σ ( (cid:96) ) = 4 × (cid:96) − for (cid:96) >
0. Every in-dividual string configuration Σ has a large effect on aspecific set of spin correlations. But by averaging overall allowed string states, the effect on a specific spin cor-relator relative to the dopant is strongly reduced.
II. MODEL
We consider a class of 2D t − J models with Hamilto-nians of the form ˆ H t − J = ˆ H t + ˆ H J , whereˆ H t = − t (cid:88) (cid:104) i , j (cid:105) (cid:88) σ ˆ P GW (cid:0) ˆ c † i ,σ ˆ c j ,σ + h.c. (cid:1) ˆ P GW (2) describes tunneling of holes with amplitude t . We con-sider fermions ˆ c i ,σ with spin σ and use Gutzwiller pro-jectors ˆ P GW to restrict ourselves to states with zero orone fermion per lattice site; (cid:104) i , j (cid:105) denotes a pair of near-est neighbor (NN) sites and every bond is counted oncein the sum. The second term, ˆ H J , includes interactionsbetween the spins ˆ S j = (cid:80) σ,τ = ↑ , ↓ ˆ c † j ,σ σ σ,τ ˆ c j ,τ with anoverall energy scale J . In the following we will considerNN Heisenberg exchange couplings,ˆ H J = J (cid:88) (cid:104) i , j (cid:105) (cid:18) ˆ S i · ˆ S j − ˆ n i ˆ n j (cid:19) (3)where ˆ n j = (cid:80) σ = ↑ , ↓ ˆ c † j ,σ ˆ c j ,σ denotes the number densityof the fermions, but the methods introduced below canbe applied more generally.To make the single-occupancy condition built into the t − J model explicit, we use a parton representation,ˆ c j ,σ = ˆ h † j ˆ f j ,σ . (4)Here ˆ h j is a bosonic chargon operator and ˆ f j ,σ is a S = 1 / (cid:88) σ ˆ f † j ,σ ˆ f j ,σ + ˆ h † j ˆ h j = 1 , ∀ j . (5)We start from the half-filled ground state | Ψ (cid:105) of theundoped spin Hamiltonian ˆ H J , and consider cases where | Ψ (cid:105) has strong AFM correlations. The ground state ofthe Heisenberg model Eq. (3) has long-range AFM or-der, but the presence of strong and short-ranged AFMcorrelations would be sufficient to justify the approxima-tions made below. We note that our results below do notrequire explicit knowledge of the wavefunction | Ψ (cid:105) .The simplest state doped with a single hole is obtainedby applying ˆ c j s ,σ to | Ψ (cid:105) , where · reverses the spin: ↑ = ↓ , ↓ = ↑ . This state, | j s , σ, (cid:105) = ˆ c j s ,σ | Ψ (cid:105) = ˆ h † j s ˆ f j s ,σ | Ψ (cid:105) , (6)with a spinon and a chargon occupying the same latticesite j s , defines the starting point for our analysis of theparton bound state constituting the magnetic polaron.In the following we assume that t (cid:29) J , which justifiesa Born-Oppenheimer ansatz: first the initially createdvalence spinon at site j s will be fixed and we determinethe fast chargon fluctuations. Similar to nuclear physics,these fluctuations can involve virtual spinon anti-spinonpairs. In a second step we introduce the trial wavefunc-tion as a superposition state of different valence spinonpositions j s . Finally, we will derive the renormalized dis-persion relation of the spinon-chargon bound state. A. Chargon fluctuations: Geometric strings andfrozen spin approximation
We review a binding mechanism of chargons andspinons by geometric strings, see also Sec. VI A in
FIG. 4.
Frozen spin approximation.
In the approxi-mate FSA basis we only allow processes where the motion ofthe chargon displaces the surrounding spins without changingtheir quantum states. As a result, nearest neighbor correla-tions C e y – red – (next-nearest neighbor correlations C e x + e y – yellow – respectively) in the frozen spin background (a) con-tribute to next-nearest neighbor correlators (nearest neighborcorrelators respectively) measured in states with longer stringlengths (b). This leads to the approximately linear string ten-sion, Eq. (10), binding spinons to chargons. Ref. [38]. When t (cid:29) J , we expect that the chargon delo-calizes until the energy cost for distorting the spin con-figuration around j s matches the kinetic energy gain. Todescribe such chargon fluctuations we apply the ”frozen-spin approximation” (FSA) [28, 44]: We assume thatthe motion of the hole merely displaces the surround-ing spins, without changing their quantum state or theirentanglement with the remaining spins. When the char-gon moves along a trajectory C starting from the spinon,it leaves behind a string Σ of displaced spins, which thuschange their corresponding lattice sites. This so-calledgeometric string is defined by removing self-retracingpaths from C . For example, the spin operator locatedat site ˜ j initially, becomes ˜ S ˜ j = ˆ S ˜ j − e x when the chargonmoves from ˜ j − e x to ˜ j along the string Σ, see Fig. 4.The geometric string construction provides the desiredgeneralization of squeezed space from 1D [25, 29–31] to2D systems: the spins are labeled by their original lat-tice sites ˜ j before the chargon is allowed to move. We callthis space, which excludes the lattice site j s where thespinon is located, the 2D squeezed space. The motion ofthe chargon along a string Σ changes the lattice geome-try: the labels ˜ j no longer correspond to the actual latticesites occupied by the spins. In particular, this changesthe connectivity of the lattice, and spins which are NNin squeezed space can become next-nearest neighbors inreal space. Hence, in this ”geometric string” formulationthe ˆ H t part of the t − J Hamiltonian is understood as in-troducing quantum fluctuations of the underlying latticegeometry. For an illustration, see Fig. 4.When t (cid:29) J , but before the Nagaoka regime is reachedaround J/t ≈ .
05, the spins in squeezed space do nothave sufficient time to adjust to the fluctuating latticegeometry introduced by the chargon motion. We notethat the shape and orientation of the geometric stringare strongly fluctuating, which leads to spatial averagingof the effects of the string on the spins in squeezed space.This averaging is very efficient because the string is in asuperposition of various possible configurations, the to- tal number of which grows exponentially with the aver-age string length. Hence the average effect on a givenspin in squeezed space is strongly reduced, which pro-vides a justification for the FSA ansatz. More technically,this means that the coupling of the fluctuating string to(para-) magnon excitations in squeezed space is weak andcan be treated perturbatively.Now we formalize our approach. When the chargonmoves along a string Σ, starting from the state | j s , σ, (cid:105) in Eq. (6), the many-body state within FSA becomes | j s , σ, Σ (cid:105) = ˆ G Σ ˆ h † j s ˆ f j s ,σ | Ψ (cid:105) . (7)Here the string operator, defined byˆ G Σ = (cid:89) (cid:104) i , j (cid:105)∈ Σ (cid:18) ˆ h † i ˆ h j (cid:88) τ = ↑ , ↓ ˆ f † j ,τ ˆ f i ,τ (cid:19) , (8)creates the geometric string by displacing the spin statesalong Σ. The product (cid:81) (cid:104) i , j (cid:105)∈ Σ is taken over all links (cid:104) i , j (cid:105) which are part of the string Σ, starting from thevalence spinon position j s .In a 2D classical N´eel state, | Ψ N0 (cid:105) = | ... ↑↓↑ ... (cid:105) , moststring states | j s , σ, Σ (cid:105) are mutually orthonormal. Spe-cific configurations, so-called Trugman loops [34], consti-tute an exception, but within an effective tight-bindingtheory it has been shown that this only causes a weakrenormalization of the spinon dispersion [38]. Since theground state | Ψ (cid:105) of the infinite 2D Heisenberg modelhas strong AFM correlations, similar to a classical N´eelstate, we expect that the assumption that string statesform an orthonormal basis remains justified. To checkthis, we calculated all such states with string lengths upto (cid:96) ≤ j s using exactdiagonalization (ED) in a 4 × |(cid:104) j s (cid:48) , σ, Σ (cid:48) | j s , σ, Σ (cid:105)| < .
06 unless Σ = Σ (cid:48) or Σ and Σ (cid:48) are related by a Trugman loop.Now we will follow the example of Rokhsar and Kivel-son: They introduced their celebrated dimer model [50]by defining a new basis which reflects the structure of thelow-energy many-body Hilbert space in a class of micro-scopic spin systems. Similarly, we will postulate in thecontext of the FSA that all string states are mutuallyorthonormal. This defines the new basis of string states | j s , σ, Σ (cid:105) which is at the heart of the FSA. Note, however,that we will return to the full physical Hilbert space ofthe original t − J model later.For a spinon with spin σ fixed at j s , the effective stringHilbert space | j s , σ, Σ (cid:105) has the structure of a Bethe lat-tice, or a Cayley graph. Its depth reflects the maximumlength of the geometric string Σ, and the branches cor-respond to different directions of the individual stringelements. The effective Hamiltonian ˆ H t eff describing thechargon motion, i.e. fluctuations of the geometric string,consists of hopping matrix elements t between neighbor-ing sites of the Bethe lattice. In addition, the J -part ˆ H J of the t − J model gives rise to a potential energy term.Within the FSA, it can be easily evaluated,ˆ H J eff = (cid:88) j s ,σ (cid:88) Σ | j s , σ, Σ (cid:105)(cid:104) j s , σ, Σ |× (cid:104) j s , σ, Σ | ˆ H J | j s , σ, Σ (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) V pot (Σ) . (9)Off-diagonal terms (cid:104) j s (cid:48) , σ, Σ (cid:48) | ˆ H J | j s , σ, Σ (cid:105) with j s (cid:48) (cid:54) = j s give rise to spinon dynamics and will be discussed be-low. By construction of the FSA, the potential V pot (Σ)only depends on the spin-spin correlation functions in theundoped ground state, C d = (cid:104) Ψ | ˆ S d · ˆ S | Ψ (cid:105) .We proceed as in the microscopic spinon-chargon the-ory of the t − J z model [38] and simplify the effectivestring Hamiltonian further by making the linear stringapproximation: We assume that the potential dependsonly on the string length (cid:96) Σ . From considering the caseof straight strings, see Fig. 4, we obtain, V pot (Σ) ≈ dEd(cid:96) (cid:96) Σ + g δ (cid:96) Σ , + µ h , (10)with a linear string tension dE/d(cid:96) = 2 J ( C e x + e y − C e x ).The last term µ h = J (1 + C e x − C e x ) corresponds toan overall energy offset; the middle term contributes onlywhen the string length is (cid:96) Σ = 0 and describes a weakspinon-chargon attraction g = − J ( C e x − C e x ). Notethat we assumed four-fold rotational symmetry of | Ψ (cid:105) ,e.g. C e x = C e y .By solving the hopping problem on the Bethe lattice inthe presence of the string potential (10) as in Refs. [32, 33,35, 38], we obtain approximations to the spinon-chargonbinding energy E FSAsc and the bound state wavefunction, | ψ FSAsc ( j s , σ ) (cid:105) = (cid:88) Σ ψ FSAΣ | j s , σ, Σ (cid:105) . (11)Recall that we applied the strong coupling approxima-tion, valid for t (cid:29) J , and fixed the valence spinon at j s . Spinon dynamics will be discussed below. FromEq. (11) the FSA string length distribution is obtained, p FSA l = (cid:80) Σ: (cid:96) Σ = l | ψ FSAΣ | , which we used in Figs. 1 - 3. B. Spinons at half filling
Now we return to the analysis of spinons which deter-mine the low-energy properties of magnetic polarons. Webriefly review fermionic spinon representations of quan-tum AFMs and their corresponding variational wavefunc-tions. They provide the starting point for formulatinga general spinon-chargon trial wavefunction in the nextsection.For concreteness we consider the 2D Heisenberg Hamil-tonian ˆ H J in Eq. (3) at half filling. Its ground statespontaneously breaks the SU(2) spin symmetry and haslong-range N´eel order [51]. The corresponding low-energy excitations – spin-1 magnons constituting the requiredGoldstone mode – are most commonly described by abosonic representation of spins, using e.g. Schwinger- orHolstein-Primakoff bosons. Recently it has been arguedthat the high-energy excitations of the AFM ground statecan be captured more accurately by a fermionic spinonrepresentation [52] however.The fermionic spinon representation which we use inEq. (4) is partly motivated by analogy with the 1D t − J model, where spinons can be understood as forminga weakly interacting Fermi sea [53–55]. On the otherhand, Marston and Affleck [56] have shown in 2D that theground state of the Heisenberg model in the large- N limitcorresponds to the fermionic π -flux, or d -wave [53], stateof spinons. For our case of interest, N = 2, the π -fluxstate is not exact, but it can be used as a starting pointfor constructing more accurate variational wavefunctions.To this end we consider a general class of fermionic spinonmean-field states | Ψ MF ( B st , Φ) (cid:105) , defined as the groundstate at half filling of the following Hamiltonian,ˆ H f, MF = − J eff (cid:88) (cid:104) i , j (cid:105) ,σ (cid:16) e iθ Φ i , j ˆ f † j ,σ ˆ f i ,σ + h.c. (cid:17) + B st (cid:88) j ,σ ( − j x + j y ˆ f † j ,σ ( − σ ˆ f j ,σ , (12)with Peierls phases θ Φ i , j = ( − j x + j y + i x + i y Φ / ± Φ per plaquetteand a staggered Zeeman splitting ∼ B st which can beused to explicitly break the SU (2) symmetry.A trial wavefunction for the SU (2) symmetric π -fluxstate is obtained by applying the Gutzwiller projection[19, 53] to the mean-field state with Φ = π and B st = 0,i.e. | Ψ π (cid:105) = ˆ P GW | Ψ MF (0 , π ) (cid:105) . Although it features nolong-range AFM order, this trial state leads to a verylow variational energy at half filling and it is also of-ten considered as a candidate state at finite doping [20].Another extreme is the Φ = 0 uniform RVB state with B st = 0, i.e. | Ψ (cid:105) = ˆ P GW | Ψ MF (0 , (cid:105) , which also yields avery good variational energy at half filling.The best variational wavefunction of the general typeˆ P GW | Ψ MF ( B st , Φ) (cid:105) has been found to have a non-zerostaggered field B st (cid:54) = 0, consistent with the broken SU (2) symmetry of the true ground state, and stag-gered flux 0 < Φ < π [57]. More recent calcu-lations determined the optimal variational parametersof this ”staggered-flux + N´eel” (SF+N) trial state | Ψ SF+N (cid:105) = ˆ P GW | Ψ MF ( B optst , Φ opt ) (cid:105) to be Φ opt ≈ . π and B optst /J eff ≈ .
44 [52]. The corresponding variationalenergy per particle E SF+N0 /L = − . J is very close tothe true ground state energy E /L = − . J knownfrom first-principle Monte-Carlo simulations [58].The main shortcoming of mean-field spinon theories asin Eq. (12) is that they neglect gauge fluctuations [49].These lead to spinon confinement in the ground state ofthe 2D Heisenberg model [49] and, hence, free spinonexcitations as described by Eq. (12) cannot exist indi-vidually. Indeed, if the Gutzwiller projection method isused to define a variational wavefunction, the underlyingmean-field spinon dispersion is usually not considered tohave a concrete physical meaning. We emphasize, how-ever, that a single spinon can exist in combination witha chargon if they form a meson. In this case, which is ofprimary interest to us, we argue that the spinon disper-sion (13) has a concrete physical meaning.The main difference between spinon models with dif-ferent values of the staggered flux Φ is their dispersionrelation. From the Hamiltonian Eq. (12) we obtain themean-field spinon dispersion ω s ( k ) = − (cid:114) J (cid:12)(cid:12)(cid:12) cos( k x ) e − i Φ4 + cos( k y ) e i Φ4 (cid:12)(cid:12)(cid:12) + B . (13)For B st = 0 and Φ (cid:54) = 0 it has Dirac points at the nodalpoint k = ( π/ , π/ B st opens a gap everywhere. In this case the dispersionhas a minimum at k = ( π/ , π/ , π ) and the nodal point ( π/ , π/
2) is zerofor Φ = 0 and maximal when Φ = π , see Fig. 5 (a).For the optimal variational parameters Φ opt and B optst ,the shape of the mean-field spinon dispersion relation(13) closely resembles the known dispersion of a singlehole moving inside an AFM: It is weakly dispersive onthe edge of the MBZ, has its minimum at ( π/ , π/
2) anda pronounced maximum at (0 , t (cid:29) J . C. Meson trial wavefunction
To obtain a complete description of the meson-likebound state constituting a hole in an AFM, we com-bine geometric strings with the fermionic spinon repre-sentation. Starting from Eq. (7) with | Ψ (cid:105) = | Ψ SF+NMF (cid:105) ≡| Ψ MF ( B optst , Φ opt ) (cid:105) we construct a translationally invari-ant trial wavefunction, | Ψ sc ( k MP ) (cid:105) = (cid:88) j s e i k MP · j s L (cid:88) Σ ψ Σ ˆ G Σ ˆ P GW ˆ f j s ,σ | Ψ SF+NMF (cid:105) . (14)Here k MP denotes the total lattice momentum of thespinon-chargon magnetic polaron state, and L denotesthe linear system size. Note that we dropped the chargonoperators ˆ h j in the expression because they are uniquelydefined by the condition Eq. (5) after the Gutzwiller pro-jection in our case with single hole.In general, the values of the string wavefunction ψ Σ ∈ C can be treated as variational parameters in Eq. (14),but in practice we use the result obtained explicitly fromthe FSA calculation in Eq. (11): I.e. we set ψ Σ inEq. (14) equal to ψ FSAΣ determined in Eq. (11). The -4.5-4-3.5-3-2.5-255.566.57-10-9.5-9-8.5
FIG. 5.
Dispersion relation of a single hole in an AFM.
We consider cuts through the Brillouin zone along the pathsketched in (a), starting at (0 , B st = 0 . J eff for Φ = 0 . π (left) and Φ = 0 , π (right). (b) We calculate themagnetic polaron dispersion for t = 3 J in a 12 ×
12 systemfrom the meson trial wavefunction, Eq. (14), (string-VMC)and compare it to predictions by the analytical FSA theorycombined with a simplified tight-binding expression for thespinon dispersion, Eq. (18). (c) As expected from the strongcoupling spinon-chargon picture, only the part of the varia-tional energy associated with spin-exchange terms (cid:104) ˆ H J (cid:105) de-pends on k MP (top), whereas (cid:104) ˆ H t (cid:105) is non-dispersive (bottom). undoped parent state | Ψ SF+NMF (cid:105) in Eq. (14) can be re-placed by any fermionic spinon mean-field wavefunction | Ψ f MF (cid:105) . In particular, this allows to use Eq. (14) to de-scribe spinon-chargon bound states even in phases withdeconfined spinon excitations.We recapitulate the physics of Eq. (14): First, the va-lence spinon is created in the mean-field state. At strongcouplings it carries the total momentum k MP of themeson-like bound state, ˆ f k MP ,σ = L − (cid:80) j e i k MP · j ˆ f j ,σ .The Gutzwiller projection subsequently yields a state inthe physical Hilbert space,ˆ P GW ˆ f k MP ,σ | Ψ SF+NMF (cid:105) = (cid:88) j s (cid:88) α Φ k MP ( j s , α )ˆ h † j s ˆ f j s ,σ | α (cid:105) (15)where (cid:80) α denotes a sum over all half-filled Fock states | α (cid:105) . In this new state the spinon and chargon positions j s coincide. In the last step we apply the string operatorsˆ G Σ to this state and create a superposition of fluctuatinggeometric strings in Eq. (14), which captures the internalstructure of the meson-like bound state. D. Simplified tight-binding description of spinons
To obtain more qualitative analytical insights to theproperties of meson-like spinon-chargon bound states, wereturn to the simplified FSA description developed inSec. II A and extend it by an approximate tight-bindingtreatment of the spinon dispersion. This approach cap-tures fewer details than the trial wavefunction Eq. (14)but provides an intuitive physical picture of the mainfeatures revealed in the spinon dispersion.So far the J -part of the effective FSA Hamiltonian,ˆ H J eff in Eq. (9), includes only the string potential. Now weadd terms J s ( j s , j s ; Σ , Σ ) | j s , σ, Σ (cid:105)(cid:104) j s , σ, Σ | changingthe position of the valence spinon, with matrix elements J s ( j s , j s ; Σ , Σ ) = (cid:104) j s , σ, Σ | ˆ H J | j s , σ, Σ (cid:105) . (16)We first evaluate Eq. (16) for string states | j s , σ, Σ (cid:105) con-structed from a classical N´eel state | Ψ (cid:105) = | Ψ N0 (cid:105) . Ignor-ing loop configurations of the strings Σ , [34], one ob-tains non-zero matrix elements J s = J/ j s and j s can be connected by two links in arbitrary directions andif the string length changes by two units, (cid:96) Σ = (cid:96) Σ ± (cid:104)(cid:104) j s , j s (cid:105)(cid:105) .Next we check that these terms remain dominant whenthe string states are constructed from the exact groundstate | Ψ (cid:105) of the 2D Heisenberg model. To this end weperformed ED in a 4 × (cid:104)(cid:104) j s , j s (cid:105)(cid:105) remain dominant with magnitudes J s = 0 . J close to the result from the classical N´eelstate. In contrast, matrix elements (16) between stateswith spinons on neighboring sites (cid:104) j s , j s (cid:105) remain small, | J s ( j s , j s ) | < . J , and will be neglected.In a generic quantum AFM we expect that this pictureremains valid, at least qualitatively. The spin-exchangecouplings on the bonds around the spinon indicated inFig. 6 can be written as ˆ S i · ˆ S j = ˆ P i , j / − /
4, whereˆ P i , j | σ i , σ j (cid:105) = | σ j , σ i (cid:105) exchanges the two spins irrespec-tive of their orientation. Hence both spins change theirsublattice index, which is expected to lead to a large over-lap with a state describing a geometric string of length (cid:96) Σ = (cid:96) Σ ± J s ≈ J/ | Ψ (cid:105) .Now we focus on strong couplings, t (cid:29) J , and makea Born-Oppenheimer ansatz [59]. As in Eq. (11) we firstfix the spinon position and determine the ground stateof the fluctuating geometric string, | ψ FSAsc ( j s , σ ) (cid:105) . Herewe have to be careful in order to avoid double counting: FIG. 6.
Simplified tight-binding description of thespinon dispersion.
In a classical N´eel state, a spinon atsite j s (left) can hop to another site j s by spin-exchange pro-cesses on the bonds indicated by ellipses. The spin-exchangeon the bond indicated by the red ellipse leads to the spinonconfiguration shown on the right, where j s = j s +2 e x and thestring length changes by two units, (cid:96) Σ = (cid:96) Σ + 2. Similarly,spinons at j s = j s − e x , ± e y , + e x ± e y and − e x ± e y canbe reached, with (cid:96) Σ = (cid:96) Σ ± In our original derivation of the FSA string potential inEq. (10) we included the energy J (cid:104) ˆ S i · ˆ S j (cid:105) between anypair of spins on sites (cid:104) i , j (cid:105) as a constant energy. However,we argued above that the exchange part J ˆ P i , j / J ˆ S i · ˆ S j on the bonds shown inFig. 6 lead to spinon dynamics, and we will include themin the tight-binding spinon Hamiltonian. Hence, to avoiddouble counting of terms in ˆ H J , we modify the effectivestring potential in Eq. (10) by subtracting J (cid:104) ˆ P i , j (cid:105) / J (cid:104) Ψ | ˆ S i · ˆ S j + / | Ψ (cid:105) for bonds (cid:104) i , j (cid:105) contributing to thematrix elements in Eq. (16).Next we include spinon dynamics. Due to the pres-ence of geometric strings, the spinon hopping elements J s between sites (cid:104)(cid:104) j s , j s (cid:105)(cid:105) are renormalized by a Franck-Condon overlap ν FC , J ∗ s = (cid:88) Σ , Σ J s ( j s , j s ; Σ , Σ ) ( ψ FSAΣ ) ∗ ψ FSAΣ = ν FC J s , (17)where ψ FSAΣ denotes the FSA string wavefunction fromEq. (11). The resulting tight-binding hopping Hamilto-nian gives rise to the strong coupling expression for thespinon-chargon energy, E sc ( k ) = 2 J s ν FC (cid:2) k x + k y ) + 2 cos( k x − k y )+ cos(2 k x ) + cos(2 k y ) (cid:3) + E FSAsc , (18)where E FSAsc is the energy contribution from the fluctu-ating geometric string.The Franck-Condon factor can be calculated in the lim-its t/J → ∞ ,
0. For weak couplings t (cid:28) J the stringlength becomes short and the Franck-Condon factor ap-proaches zero, ν FC →
0. This leads to a strong suppres-sion of the magnetic polaron bandwidth W . For strongcouplings, t (cid:29) J , the Franck-Condon factor approaches ν FC = 1 /
2. This leads to a bandwidth W ∝ J (cid:28) t .0 III. RESULTS
Now we present numerical results from the micro-scopic spinon-chargon theory, obtained from the simpli-fied tight-binding description, see Sec. II D, and from thevariational energy of the trial wavefunction in Eq. (14).To calculate the latter we utilize standard Metropo-lis sampling, commonly employed in variational MonteCarlo (VMC) calculations, see e.g. Ref. [60].
A. Dispersion relation
Shape.–
In Fig. 5 (b) we show the variational dispersionrelation, E MP ( k MP ) = E ( k MP ) − E of the single holein the AFM, or magnetic polaron, where E = E SF+N0 isthe variational energy of the SF+N state without dopingand E ( k MP ) = (cid:104) Ψ sc ( k MP ) | ˆ H t − J | Ψ sc ( k MP ) (cid:105) for dopingwith one hole. The result is in good quantitative agree-ment with numerical Monte Carlo calculations [12, 16],capturing all properties of the single-hole dispersion.From the spinon-chargon theory we expect that thedispersion of a hole in an AFM is dominated by thespinon properties when t (cid:29) J . Indeed, the shape ofthe variational dispersion in Fig. 5 (b) closely resemblesthe mean-field spinon dispersion, see Fig. 5 (a). To cor-roborate this picture further, we calculate the variationalenergies (cid:104) ˆ H J (cid:105) and (cid:104) ˆ H t (cid:105) as a function of k MP individuallyin Fig. 5 (c). Only the spin-exchange part (cid:104) ˆ H J (cid:105) is disper-sive, whereas the chargon part (cid:104) ˆ H t (cid:105) does not depend on k MP within error bars. This is a direct indication that ahole in an AFM has two constituents, one of which mainlyaffects spin exchanges and determines the dispersion ofthe meson-like bound state.In Fig. 5 (b) we also compare our result from thespinon-chargon trial wavefunction to the FSA tight-binding prediction from Eq. (18). While there is remark-able overall agreement, the semi-analytical FSA tight-binding calculation misses some important qualitativefeatures: Eq. (18) does not capture the minimum of thesingle-hole dispersion at ( π/ , π/
2) but predicts a degen-erate minimum along the edge of the MBZ, resemblingmore closely the mean-field spinon dispersion (13) with-out staggered flux, Φ = 0, see Fig. 5 (a).
Bandwidth.–
In Fig. 7 we vary the ratio
J/t and calcu-late the bandwidth W = E MP (0 , − E MP ( π/ , π/ t (cid:29) J , our results from the trial wave-function (14) (string-VMC) are in good agreement withnumerical results from various theoretical approaches.When t (cid:38) J/
3, the errorbars of our variational results arelarge. In this regime the average length of the geometricstring exceeds one lattice site, and our string-VMC calcu-lations suffer from strongly fluctuating numerical signs.We also compare the bandwidth W of the magnetic po-laron to the tight-binding prediction in Eq. (18). Whilequantitative agreement is not achieved everywhere, theoverall dependence on J/t is accurately captured. At -1 FIG. 7.
Bandwidth W of a single hole in an AFM. Wecompare our variational results (string-VMC) from Eq. (14),valid at strong couplings, to quantum Monte Carlo simula-tions by Brunner et al. [16], ED studies by Dagotto et al. [10]and Leung and Gooding [15], a variational wavefunction bySachdev [8], and spin-wave calculations by Martinez et al.[11]and by Liu and Manousakis [12]. The overall shape of W ( J/t )is well captured by the effective FSA theory, Eq. (18). Thestring-VMC calculations are performed at B st = 0 . J eff ,Φ = 0 . π in a 12 ×
12 system. strong couplings, t (cid:29) J , the bandwidth W ∝ J is pro-portional to J , i.e. strongly suppressed relative to t . Atweak couplings, t (cid:28) J , the Franck-Condon factor van-ishes, ν FC →
0, which leads to a strong suppression of W relative to both t and J . B. Ground state energy
In Fig. 8 we calculate the ground state energy of themeson-like bound state as a function of
J/t . The vari-ational result (string-VMC) from the trial wavefunction(14) agrees well with numerically exact Monte Carlo cal-culations [17] at strong couplings, before the fluctuatingsigns prevents efficient numerical calculation. The devi-ations from the exact result are on the order of J evenwhen J > t . When t (cid:29) J , the ground state energy of thesingle hole is of the form E MP = − √ t + c t / J / + O ( J ), which can be understood as a consequence of thegeometric string with an approximately linear string ten-sion [32, 38]. The variational ground state energy is ac-curately described by the semi-analytical FSA predictionfrom Eq. (18), for all values of t/J .So far we evaluated the spinon-chargon trial wave-function (14) using the FSA string wavefunction and set ψ Σ = ψ FSAΣ , where ψ FSAΣ was obtained from Eq. (11).Because the number of allowed string states Σ growsexponentially with the maximum length of the strings,it is numerically too costly to treat all amplitudes ψ Σ as variational parameters. To study the quality of thetrial state, we now introduce a single variational param-eter. We calculate the string wavefunction ψ Σ in theover-complete FSA Hilbert space but modify the po-tential in Eq. (10) by rescaling the linear string ten-sion dE/d(cid:96) → λ dE/d(cid:96) dE/d(cid:96) . The numerical factor1 FIG. 8.
Ground state energy of a hole in an AFM.
Wecompare our variational result from the meson trial wavefunc-tion in Eq. (14) (string-VMC) to quantum Monte Carlo calcu-lations by Mishchenko et al. [17] (QMC) and semi-analyticalpredictions by the tight-binding FSA theory from Eq. (18). Inthe shaded region, defined by
J < . t , the ground state isexpected to be a Nagaoka polaron [18, 39]. The string-VMCcalculations are performed at B st = 0 . J eff , Φ = 0 . π in a12 ×
12 system at k MP = ( π/ , π/ λ dE/d(cid:96) ≥ λ dE/d(cid:96) . We observe a minimum at approximately λ dE/d(cid:96) ≈
4, around which the variational energy de-pends rather insensitively on λ dE/d(cid:96) . For larger values of λ dE/d(cid:96) , where the average length of the geometric stringis close to zero, higher variational energies are obtained.This indicates that the formation of geometric strings isenergetically favorable. For smaller values of λ dE/d(cid:96) theaverage string length exceeds one lattice constant. Thismakes the fluctuations of the numerical sign in the VMCmethod worse, but our results indicate an increase of thevariational energy in this regime as well. In combination,these results support the spinon-chargon pairing mecha-nism by geometric strings. C. Magnetic polaron cloud
Finally we study the dressing cloud of magnetic po-larons and calculate local spin correlations from the trialwavefunction (14). In Fig. 10 (a) we compare the vari-ational result to our DMRG simulations described inSec. I C for t = 3 J . The DMRG results are based on thesame snapshots from which we obtained the string lengthhistogram in Fig. 1 (c). This method also allows us tocompare to predictions by the FSA, see Fig. 2, where ge-ometric strings are included by hand into snapshots ofan undoped Heisenberg model, see Sec. I C.For t = 3 J , i.e. for strong couplings, we find excellentagreement of the trial wavefunction with DMRG simula- -2-1.8-1.6-1.4-1.2-1-0.8-0.6 FIG. 9.
Optimization of the trial wavefunction.
Wechange the average length of the geometric string in thespinon-chargon wavefunction Eq. (14), (cid:104) (cid:96) (cid:105) = (cid:80) Σ | ψ Σ | (cid:96) Σ shown in the inset, by rescaling the linear string tension inEq. (10) with a factor λ dE/d(cid:96) . The resulting variational en-ergy (cid:104) ˆ H t − J (cid:105) − E , in units of J , is calculated as a functionof λ dE/d(cid:96) for the parameter t = 2 J . The string-VMC calcula-tions are performed at B st = 0 . J eff , Φ = 0 . π in a 12 × k MP = ( π/ , π/ tions. The numerical results confirm that diagonal next-nearest neighbor correlations next to the mobile hole arestrongly suppressed, i.e. C (1 / √ ≈
0. At distances d (cid:38) .
5, no significant dependence of the correlations on t/J can be identified by either method.The most striking feature predicted by DMRG and thetrial wavefunction is the formation of a peak with reducednearest neighbor correlations C ( d ) at the distance d =2 .
06 from the mobile dopant. This feature becomes morepronounced as t/J increases. Additionally, we observeenhanced nearest neighbor correlations C ( d ) at d = 1 . t/J (cid:38) .
5, seeFig. 11 (a) and (b).The additional spatial structure featured by the trialwavefunction and DMRG at d = 1 . d = 2 .
06 is notcaptured by the simplified FSA approach, see Fig. 11 (c).The fact that the feature is present in the trial wavefunc-tion when t (cid:28) J indicates that it is caused by the mi-croscopic correlations of the spinon position with its spinenvironment. The emergence of a second length scale,in addition to the string length (cid:96) ∝ ( t/J ) / capturedby the FSA, can be considered as an indirect indicationof fermionic spinon statistics: The Fermi momentum k F defines a second intrinsic length scale in this case.For weaker couplings, t (cid:46) J , the trial wavefunction isless accurate since the Born-Oppenheimer approximationis no longer valid. As shown in Fig. 11 (a), it predicts astrong suppression of diagonal next-nearest neighbor cor-relations around the hole, C (1 / √ ≈
0, for all valuesof t/J . In contrast, the DMRG features a strong depen-dence of C (1 / √
2) on t/J when t (cid:46) J , which is accu-rately described by the FSA, see Fig. 11 (b) and (c). Onthe other hand, the FSA approach is less reliable for the2nearest neighbor correlations, whose qualitative shape isremarkably well described by the trial wavefunction forall values of t/J .In Fig. 10 (b) we calculate local spin-spin correlationsat a distance d from the hole, for different total momenta k of the spinon-chargon bound state. We observe thatthe feature in C ( d ) at d = 2 .
06 is most pronouncedin the ground state at k = ( π/ , π/ k = (0 , FIG. 10.
Magnetic polaron cloud.
Using the trial wave-function (string-VMC) in Eq. (14) we calculate the localspin correlations C n ( d ) = (cid:104) ˆ n h r h ˆ S r · ˆ S r (cid:105) / (cid:104) ˆ n h r h (cid:105) , where d = | ( r + r ) / − r h | is the bond-center distance between r and r and n = 1 ( n = 2) corresponds to nearest (next-nearest)neighbor spin correlations. (a) For t = 3 J we compare ourstring-VMC result C n ( d ) / C zn ( d ) obtained from the sameDMRG simulations as in Fig. 2: MPS corresponds to the fullsolution in the case of a mobile hole and the FSA predic-tions are generated from snapshots of the undoped Heisen-berg AFM. (b) We calculate the momentum dependence of C n ( d ) from the trial wavefunction at t = 2 J . The string-VMCcalculations in (a) and (b) are performed at B st = 0 . J eff ,Φ = 0 . π in a 14 ×
14 system; in (a) k = ( π/ , π/ characteristic structure of C ( d ) is related to the spinon,which also carries the center-of-mass momentum k of themagnetic polaron at strong couplings. IV. DISCUSSION AND OUTLOOK
We have introduced a microscopic theoretical frame-work to describe a hole doped into an AFM as a meson-like bound state of a spinon and a chargon. As a bindingmechanism of spinons and chargons at strong couplings,we suggest geometric strings: They model how the char-gon motion dynamically changes the underlying latticegeometry. The trial wavefunction introduced in this ar-ticle puts earlier results [40] on a mathematical footing,allowing quantitative predictions beyond the simplified t − J z model [38]. While we focus on a single hole atzero temperature in a spin system with long-range AFMorder, our method should also be applicable at finite dop-ing, for higher temperatures and in systems without long-range N´eel order.Our results obtained here for the energy, the disper-sion relation and the magnetic dressing cloud of a singlehole in an AFM are in good agreement with the com-monly used magnetic polaron theory [6–8, 11, 12]. Thespinon-chargon approach can be understood as a refine-ment of the magnetic polaron picture. We find that manyproperties of the polarons formed by holes in the FermiHubbard model follow more directly from the spinon-chargon ansatz. Moreover, simple theoretical picturescan be derived from the parton approach at strong cou-plings, where the magnetic polaron theory becomes no-toriously difficult to solve, and our approach is particu-larly useful for understanding the structure of magneticpolarons in real space, which has recently become acces-sible by quantum gas microscopy of ultracold atoms inoptical lattices [47].Another key advantage of the spinon-chargon approachis that it continuously connects systems with and withoutlong-range AFM order. In contrast to the magnetic po-laron theory, our variational wavefunction captures cor-rectly the physics of the 1D t − J model, where geometricstrings become infinitely long and spinons are no longerbound to chargons. We thus believe that our method iswell suited to study the dimensional cross-over from the1D to the 2D t − J model in the future. In this articlewe only considered the case when spinons and chargonsform a bound state, but we do not exclude the possibilitythat the attractive potential between spinons and char-gons is finite and an unbound state could exist at finiteenergy. This would correspond to a phase with decon-fined spinons.As an important application, we expect that our ap-proach can also provide the means for a microscopic de-scription, starting form first principles, of the FL ∗ stateproposed as an explanation of the pseudogap phase incuprates [61]. As a key ingredient, the FL ∗ state con-tains bound states of spinons and chargons, similar to3 t=0.01 Jt=0.11 Jt=0.4 Jt=0.9 Jt=1.7 Jt=3 J t=0.01 Jt=0.11 Jt=0.4 Jt=0.9 Jt=1.7 Jt=3.0 J t=0.01 Jt=0.11 Jt=0.4 Jt=0.9 Jt=1.7 Jt=3 J FIG. 11.
Comparison of the different theoretical approaches.
We calculate the local spin correlations C n ( d ) / C zn ( d )for n = 1 , B st = 0 . J eff , Φ = 0 . π in a14 ×
14 system at k = ( π/ , π/ the meson-like bound states discussed in this paper. Wepropose geometric strings as a possible spinon-chargonbinding mechanism in this finite-doping regime. Indeed,recent experiments in the corresponding region of thecuprate phase diagram [28] have found indications forthe presence of geometric strings, and here we confirmedthese results at low doping and for zero temperature bystate-of-the-art DMRG simulations.To shed more light on the connection between thespinon-chargon trial wavefunction and the pseudogapphase, as a next step it would be useful to study spectralproperties of mesons as measured in angle-resolved photoemission spectroscopy (ARPES) experiments. From thestrong coupling wavefunction introduced here, we expecttwo contributions to the spectral weight: a spinon part,which is strongly dispersive, and a chargon, or string,contribution which only has a weak momentum depen-dence. We expect that this allows to draw further analo-gies with ARPES spectra in 1D systems [54, 55], and itmay provide new insights to the physics of Fermi arcsobserved in the pseudogap phase of cuprates.We close by a comment about the relation of ourapproach to Anderson’s resonating valence bond pictureof high-temperature superconductivity [19, 62]. As inhis approach, we use Gutzwiller projected mean-fieldstates of spinons [53] as key ingredients in our trialwavefunction. By adding geometric strings we includeshort-range hidden order and take Anderson’s ansatz ina new direction. Our method is not based on spin-chargeseparation but instead describes meson-like boundstates of spinons and chargons. The implications forunconventional superconductivity will be explored in thefuture. ACKNOWLEDGEMENTS
We would like to thank E. Altman, M. Knap, M. Punk,S. Sachdev, T. Shi, R. Verresen, Y. Wang and Z. Zhufor useful feedback and comments. We also acknowledge fruitful discussions with I. Bloch, C. Chiu, D. Chowd-hury, D. Greif, M. Greiner, C. Gross, T. Hilker, S. Huber,G. Ji, J. Koepsell, S. Manousakis, F. Pollmann, A. Rosch,G. Salomon, U. Schollw¨ock, L. Vidmar, J. Vijayan andM. Xu.F.G. acknowledges support by the Gordon and BettyMoore foundation under the EPIQS program. F.G. andA.B. acknowledge support from the Technical Universityof Munich - Institute for Advanced Study, funded by theGerman Excellence Initiative and the European UnionFP7 under grant agreement 291763, from the DFG grantNo. KN 1254/1-1, and DFG TRR80 (Project F8). A.B.also acknowledges support from the Studienstiftung desdeutschen Volkes. E.D. and F.G. acknowledge supportfrom Harvard-MIT CUA, NSF Grant No. DMR-1308435,AFOSR Quantum Simulation MURI.4
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