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Quantum Monte Carlo methods for charged systems Charged systems are the basic model of condensed matter physics How can this be done with cold atoms? ÞHow can they be simulated with Quantum Monte Carlo? Phase diagram of the “one component plasma” in 2D (as important as hubbard model?)

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Many different quantum Monte Carlo methods Which ensemble? T=0 VMC (variational) DMC/GFMC (projector) or T>0? PIMC (path integrals) Which basis? Particle Coordinate Space S z representation for spin Occupation Number Lattice models Wave functions Hartree-Fock Slater-Jastrow Backflow, 3 body Localized orbitals in crystal Which Hamiltonian? Continuum Lattice (Hubbard, LGT)

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Imaginary-time path integrals The thermal density matrix is: Trotter’s theorem (1959):

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“ Distinguishable” particles within PIMC Each particle is a ring polymer; an exact representation of a quantum wavepacket in imaginary time. Integrate over all paths The dots represent the “start” of the path. (but all points are equivalent) The lower the real temperature, the longer the “string” and the more spread out the wavepacket. Path Integral methods can calculate all equilibrium properties without uncontrolled approximations We can do ~2000 charges with ~1000 time slices.

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Bose/Fermi Statistics in PIMC Average by sampling over all paths and over connections. At the superfluid transition a “macroscopic” permutation appears. This is reflection of bose condensation within PIMC. Fermion sign problem: -1 for odd permutations.

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Projector Monte Carlo (T=0) aka Green’s function MC, Diffusion MC Automatic way to get better wavefunctions. Project single state using the Hamiltonian This is a diffusion + branching operator. Very scalable: each walker gets a processor. This a probability for bosons/boltzmanons since ground state can be made real and non-negative. Use a trial wavefunction to control fluctuations and guide the random walk; “importance sampling” For a liquid we use a Jastrow wavefunction, for a solid we also use Wannier functions (Gaussians) to tie particles to lattice sites. More accurate than PIMC but potentially more biased by the trial wavefunction.

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How can we handle charged systems? If we cutoff potential : –Effect of discontinuity never disappears: (1/r) (r 2 ) gets bigger. –Will not give proper plasmons because Poisson equation is not satisfied Image potential solves this: V I = v(r i -r j +nL) –But summation diverges. We need to resum. This gives the Ewald image potential. –For one component system we have to add a background to make it neutral (background comes from other physics) –Even the trial function is long ranged and needs to be resummed.

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Classic Ewald Split up using Gaussian charge distribution If we make it large enough we can use the minimum image potential in r-space. Extra term for insulators:

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9 Computational effort r-space part same as short-ranged potential k-space part: 1.Compute exp(ik 0 x i ) =(cos (ik 0 x i ), sin (ik 0 x i )), k 0 =2 /L. 2.Compute powers exp(i2k 0 x i ) = exp(ik 0 x i )*exp(ik 0 x i ) etc. Get all values of exp(ik. r i ) with just multiplications. 3.Sum over particles to get k for all k. 4.Sum over k to get the potentials. Constant terms to be added. Table driven method used on lattices is O(N 2 ). For N>1000 faster methods are known. O(N 3/2 ) O(N) O(N 3/2 ) O(N 1/2 ) O(1)

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Fixed-node method Impose the condition: This is the fixed-node BC Will give an upper bound to the exact energy, the best upper bound consistent with the FNBC. f(R,t) has a discontinuous gradient at the nodal location. Accurate method because Bose correlations are done exactly. Scales well, like the VMC method, as N 3. Can be generalized from the continuum to lattice finite temperature, magnetic fields, …

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Dependence of energy on wavefunction 3d Electron fluid at a density r s =10 Kwon, Ceperley, Martin, Phys. Rev. B58,6800, 1998 Wavefunctions –Slater-Jastrow (SJ) –three-body (3) –backflow (BF) –fixed-node (FN) Energy converges to ground state Variance to zero. Using 3B-BF gains a factor of 4. Using DMC gains a factor of 4. FN -SJ FN-BF

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One component system of charges: dimensionless parameters 3d: electrons in metals C nuclei in white dwarfs 2d: electrons/ions on surfaces charged colloids on a surface 1d: electrons in wires 0d: electrons in dots Large r s = strong correlation, Wigner crystal

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The 2D one component plasma PRL 103, 055701 (2009); arXiv:0905.4515 (2009) Bryan Clark, UIUC & Princeton Michele Casula, UIUC & Saclay, France DMC UIUC Support from: NSF-DMR 0404853 Electrons or ions on liquid helium Semiconductor MOSFET charged colloids on surfaces We need cleaner experimental systems!

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Phase Diagram for 2d boson OCP (up to now) Wigner crystal Classical plasma Quantum fluid 1 / r s =(density) 1/2 T ( R) Quantum- classical crossover Hexatic phase ~124 r s ~ 60 ~ 3x10 12 cm 2 Electrons on helium electrons De Palo, Conti, Moroni, PRB 2004.

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Kosterlitz-Thouless transition Mermin-Wagner theorem implies that a finite temperature crystal is unstable KTHNY showed that an instability in the solid comes from dislocation unbinding. There should be a hexatic phase sandwiched between liquid and crystal for T>0 Melting occurs via 2 second order transitions. Classical OCP makes a transition at 122< <124.

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Inhomogenous phases Cannot have 2-phase coexistence at first order transition! the background forbids it Jamei, Kivelson and Spivak [Phys. Rev. Lett. 94, 056805 (2005)] “proved” (with mean field techniques) that a 2d charged system cannot make a direct transition from crystal to liquid a stripe phase between liquid and crystal has lower energy Does not prove that stripes are the lowest energy state, only that the pure liquid or crystal is unstable at the transition assume Boltzmann statistics – no fermion sign problem. F area Maxwell construction

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snapshots 122< <124 classical quantum Triangular lattice forms spontaneously in PIMC

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Structure Factors PIMC Exper. Keim, Maret, von Grunberg. Classical MC

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Hexatic order He,Cui,Ma,Liu,Zou PRB 68,195104 (2003). Muto, Aoki PRB 59, 14911(1999) r r

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g 6 (r)~r - We find that liquid-hexatic transition happens when = 2 Different than a short range system! = 1/4 “nose” region crystal T T TT All liquid simulations

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2d OCP Phase Diagram PRL 102,055701 (2009) Wigner crystal Classical plasma Quantum fluid 1 / r s T ( R) Hexatic phase Clausius-Clapeyron relation : First order transition is on “nose”

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Transition order differs from KT? 1 / r s T ( R) 1 st order 2 nd order? Internal energy vs T

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Ground state fluid-solid transition We have established a much more accurate estimation of the transition at rs=66.5 Careful estimation of finite size errors using theory of Chiesa et al. Phys. Rev. Letts. 97, 076404:1-4 (2006); Metastable points

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Where to look for inhomogenous phases? Wigner crystal Classical plasma Quantum fluid 1 / r s T ( R) Quantum- transition Hexatic phase nothing

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Unusual structure in peak of S(k) Could be caused by many small crystals r s =65, N=2248

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Structures exist in transition region Are structures real? Or an ergodic problem Are they different from a liquid? Can we make a ground state model? Not one that is energetically favorable.

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T=0 calculation of stripes Energy No evidence for stripes at T=0 Stripe width~ 10 500 a liquid crystal

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Is there a dipole layer on the stripe? Absence of dipole layer invalidates the “theorem.” DMC calculation at rs=60 If there is a dipole layer it is quite small. Is a dipole layer possible in a one-component charged system?

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Point defects in the 2D Wigner crystal Candido et al. Phys. Rev. Lett. 86, 492-495 (2001) Add and subtract an electron, keeping density fixed. Calculate total energies using path integrals Interstitials have a lower energy than vacancies! Cockayne-Elser harmonic energy Anharmonic terms reduce the defect energy for r s ~100 The creation energy for an interstitial vanishes for r s ~35; very close to the melting density vacancy interstitial

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Wigner crystal Normal fluid superfluid hexatic 2D Bose OCP T ( R)

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2dOCP fermion Phase diagram 2d Wigner crystal is a spin liquid. Magnetic properties are nearly divergent at melting (2d) and (nearly) 2 nd order melting. But sign problem? UNKNOWN Quantum Fluid Super- conductor?

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Phase Diagram of 3DEG Polarization transition second order partially polarized transition at r s =52 like the Stoner model (replace interaction with a contact potential) Antiferromagnetic Wigner Crystal at r s >105

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Magnetism in the 2DWC Bernu et al PRL 86, 870-873 (2001). Melting line Regime of strong quantum effects Regime of magnetic effects Potential energy dominates r s =35 T=0 melting

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Conclusions Long-ranged interactions are not an intractable problem for simulation. We have established the outlines of the OCP phase diagram for boltzmannons and bosons. Evidence for intervening inhomogeneous phases is weak Future work: fermi statistics – but the “sign problem” makes the fluid phases challenging (not hopeless). The OCP is a good target for a “quantum emulator.” With optical lattice+disorder one can reach some of the most important problems in CMP.

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