part1: non$fermiliquidphases** · pdf file · 2012-02-23part1:...

Part 1: Non-‐Fermi Liquid phases for 2d i5nerant electrons

MPA Fisher with Hongchen Jiang, MaA Block, Ryan Mishmash, Donna Sheng, Lesik Motrunich (in progress)

•  Wavefunc5ons for NFL – partons •  Example of a NFL: “D-‐wave Metal” •  Hamiltonian and solu5on with DMRG

Goal: Construct and Analyze non-‐Fermi liquid phases of strongly interac5ng 2d i5nerant electrons

Simons Symposia, Quantum Physics Beyond simple systems US Virgin Islands, Jan 31, 2012

2D Free Fermi Gas

Volume of Fermi sea set by par5cle density

Momentum Distribu5on Func5on:

k kF

1

⇥ = k2F /4�

2D Fermi Liquid

k kF

1

Z < 1

hc†kcki

hc†kcki

Lu_ngers Thm: Volume inside Fermi surface s5ll set by total density of fermions

kx

ky

⇥ = k2F /4�

2D Non-‐Fermi Liquid?

k kF

Various possibili5es:

1)  A singular “Fermi surface” that sa5sfies Lu_nger’s theorem but no jump discon5nuity 2) A singular Fermi surface that violates Lu_nger’s theorem 3) A singular “Fermi surface” with ``arc”

4.) Other….

Wavefunc5ons for 2d Fermions

Free Fermi gas: Slater determinant �FF (ri�, ri⇥) = det[eiki·rj")] det[eiki·rj#)]

Interac0ng Fermi liquid:

Mul5ply by Jastrow factor

�FL = e�P

i<j u(Ri�Rj) ⇥�FF {Ri} = {ri", ri#}

“Parton” approach to wavefunc5ons

Mean Field theory: treat “Spinons” and Bosons as Independent:

Decompose electron: spinless charge e boson, s=1/2 neutral fermionic spinon

Wavefunc5ons

“Fix-‐up” Mean Field Theory and project into physical Hilbert space “Glue” together Fermion and Boson “partons”

(enlarged Hilbert space -‐ twice as many par5cles)

�f (ri", ri#) b(Ri)

� ⌘ f (ri�)⇥ b(Ri ! ri�)

c� = bf�

Fermi and Non-‐Fermi Liquids?

Fermi Liquid: Bosons into Bose condensate

Non-‐Fermi Liquid: Bosons into uncondensed fluid -‐ a “Bose metal”

NFL Metal: Product of Fermi sea and uncondensed Bose-‐Metal

Spinons in a filled Fermi sea �f = det[eiki·rj")] det[eiki·rj#)]

�BECb = e�

Pi<j u(Ri�Rj)

c� = hbif� ⇠ (const)f�

�FL = �FFf � �BEC

b

�NFL = �FFf � �BoseMetal

b

Bose-‐metal: Fragment the charge sector

b = d1d2 (c↵ = f↵d1d2)(all Fermionic decomposi5on of the electron)

�b = det1 ⇥ det2Gutzwiller wavefunc5on:

Wavefunc5on for D-‐wave Bose-‐Metal (DBM)

DBM: Product of 2 Fermi sea determinants, elongated in the x or y direc5ons

�DBM

= detx

⇥ dety

+

+

-‐

-‐

Dxy rela5ve 2-‐par5cle correla5ons

Ques5on by Ady Stern: When you break the charge e boson into two fermionic partons, d1 and d2, how do you divide the charge? Answer by MPAF: It doesn’t maAer. Within the Gutzwiller projec5on approach to construct varia5onal wavefunc5ons, the coordinates of the d1 and d2 fermions are equated – each site is either occupied by both a d1 and a d2 fermion, or is empty. If one implements the projec5on by introducing a U(1) gauge field, the fermionic partons are not gauge invariant and their electric charge can be switched back and forth by shining the gauge field, leaving the physical (gauge invariant) boson charge unchanged.

Ques5on by Michael Freedman: In the nodal structure for your d-‐wave metal wavefunc5on, why are there two sets of nodal lines emana5ng from each par5cle? Answer by MPAF: A fermionic Slater determinant must vanish whenever the posi5ons of two fermions coincide, so that a single nodal line must pass through the loca5on of each par5cle in the 2d projec5on. For the d-‐wave Bose metal wavefunc5on, which is a product of 2 Slater determinants, there are 2 such nodal lines, one for each determinant.

“D-‐Wave Metal”

I5nerant NFL phase of 2d electrons?

Parton construc5on

Wavefunc5on; Product of determinants

Filled Fermi sea

�Metal

d

xy

= detx

[eiKi·Rj ] · dety

[eiKi·Rj ]� det[eiki·rj" ] · det[eiki·rj# ]

x y

Can use Varia5onal Monte Carlo to extract equal 5me correla5on func5ons from wf But what about energe5cs???

Ques5on by Leonid Glazman: How do you iden5fy the coordinates of the fermionic spinons with the coordinates of the d1 and d2 partons? Answer by MPAF: Picking an arbitrary ordering of the N spinons and of the N d1 and d2 partons, one simply requires that the posi5on of the ith spinon coincides with the posi5ons of the ith d1 and d2 fermions, for all i between 1 and N.

Hamiltonian with D-‐wave Metal ground state???

t-‐J-‐K “Ring” Hamiltonian

Electron singlet pair “rota5on” term

3 4

2 1

3 4

2 1

Strong coupling limit of parton gauge theory c� = f�dxdy

H = HtJ +HK

HtJ = �t�

�ij⇥

[c†i�cj� + h.c.] + J�

�ij⇥

⌥Si · ⌥Sj

HK = K�

�1234⇥

[S†13S24 + h.c.]

Phase diagram? Severe sign problem

Solu0on: Analyze on the 2-‐leg ladder

Electron t-‐J-‐K model on 2-‐leg ladder

Three parameters: J/t, K/t and density n (n=1/3 henceforth) No double occupancy

AAacked model using: ED DMRG VMC Bosoniza5on of Quasi-‐1d U(1) gauge theory

Hongchen Jiang, MaA Block, Ryan Mishmash, Donna Sheng, Lesik Motrunich and MPAF (in progress)

H = HtJ + K�

�1234⇥

[S†13S24 + h.c.]

J

Ques5on by Andy Millis: Why is it easier to study the t-‐J-‐K Hamiltonian with 1/3 electron per site than some higher density, say 2/3, per site? Answer by MPAF: At 1/3 density, the electrons par5ally fill only the bonding band – the an5bonding band is completely empty. At higher density one will (typically) have par5al filling of both bonding and an5-‐bonding bands, implying that there are more low lying excita5ons and that the system is more entangled. The numerical technique DMRG works best for quantum states that have low entanglement, which is the case at lower densi5es such as 1/3.

DMRG Phase Diagram

2-‐leg ladder at n=1/3

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

Fully

Polarized

Phase

Separation

non-Fermi

Liquid

Luttinger

Liquid

J

K

tp=t, ρ=1/3

Three Phases: (a) A Lu_nger Liquid (a “1d Fermi liquid”) (b) Fully polarized: Ferromagne5c metal (c) A Non-‐Fermi liquid: A D-‐wave Metal??

Satisfies Luttinger’s Theorem: the volume enclosed by the “Fermi surface” yields the particle density. (16 particles, singlet, 8 up and 8 down)"

Lu_nger Liquid

A canonical (single band) Lu_nger liquid

Non-‐Fermi-‐Liquid Phase

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Electron Momentum Distribution Function: K = 2.0

momentum kx

(

2⇡

48 )

nk

(k x

,ky

)

DMRG

0.0

0.2

0.4

0.6

0.8

-24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24

0.0

0.2

0.4

0.6

0.8

qy

= 0

qy

= ⇡

Strange non-‐monotonic momentum distribu5on func5on, No Lu_ngers volume apparent

Non-‐Fermi liquid Phase

Can we understand in terms of d-‐wave metal???

The d-‐wave Metal on 2 Legs

Parton construc5on: Product of Slater determinants for dx , dy and f

c� = dxdyf�

Electron momentum distribu5on func5on

b = dx

dy

Mean Field: convolu5on of partons

c� = dxdyf�

Gauge theory -‐ certain wavevectors enhanced

Illustrate with Boson ring model (MFT)

Sharp peaks in the exact boson momentum distribu5on func5on! (from DMRG)

nMFTc (k) = nd

x

(k)⌦ ndy

(k)⌦ nf (k)

nMFTb (k) = nd

x

(k)⌦ ndy

(k)

nb(k)

Ques5on by Bert Halperin: How can one understand at which wavevectors the electron momentum distribu5on func5on will have dominant singulari5es? Answer by MPAF: The singulari5es in the electron momentum distribu5on func5on occur at sums and differences of the Fermi wavevectors of the three cons5tuent partons. The dominant singulari5es are determined by Amperes law – when the gauge currents of the three partons are parallel, they are aAracted to one another so that they can propagate together effec5vely as an almost free electron with a concomitant sharp singularity in the electron momentum distribu5on func5on.

Momentum distribu5on func5on in the d-‐wave metal?

⇡�⇡ ⇡

kx

nb(k) nb(k)

nc(k) ⇡ nb(k)⌦ nf (k)?

nf (k) = �(KfF � |k|) (Free spinon sea)

Ques5on by Bert Halperin: How are the Fermi wavevectors of the partons that enter into the Gutzwiller wavefunc5on determined? Answer by MPAF: They are taken as varia5onal parameters, and chosen to minimize the energy of the Hamiltonian.

Convolu5on: c = b f

0 �⇡ ⇡k

x

nc(k) ⇡ ⌦0 �⇡ ⇡

1

kfF

nf (k)

0

nc

(kx

, ky

= 0)

kx

0 kx

nc

(kx

, ky

= �)

nb(k)

Non-‐Fermi Liquid – consistent with d-‐wave Metal!

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Electron Momentum Distribution Function: K = 2.0

momentum kx

(

2⇡

48 )

nk

(k x

,ky

)

DMRG

0.0

0.2

0.4

0.6

0.8

-24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24

0.0

0.2

0.4

0.6

0.8

qy

= 0

qy

= ⇡

Ques5on by ??: Are the singulari5es seen in the electron momentum distribu5on func5on in the figure on slide 24 jump discon5nui5es? Answer by MPAF: Almost certainly not. We expect that the form of these singulari5es is determined by Lu_nger liquid parameters which encode the interac5ons between the partons within a bosonized descrip5on of the quasi-‐1d gauge theory.

Varia5onal Monte Carlo (VMC)

�Metal

d

xy

= detx

[eiKi·Rj ] · dety

[eiKi·Rj ]� det[eiki·rj" ] · det[eiki·rj# ]

D-‐wave Metal: Product of Slater determinants (on 2-‐leg ladder)

Varia5onal Parameters: Distribu5on of dx partons between bonding/an5-‐bonding bands (f-‐spinons and dy partons only in bonding band)

Momentum Distribu5on func5on from VMC

VMC vs. DMRG

Conclusions

•  Constructed generalized t-‐J-‐K Hamiltonian, has NFL phase on 2-‐leg ladder • Varia5onal Monte Carlo shows this phase is the “d-‐wave Metal”, which one can understand with fermionic partons

Future

•  Mul5-‐leg ladders, other densi5es, and towards 2d •  VMC energe5cs on 2d t-‐J-‐K Hamiltonian (FL, D-‐wave BCS, D-‐wave Metal,…) •  Other wfs/Hamiltonians for 2d NFL phases??

Andy Millis commented that for a free fermion determinant it is believed that there are “pockets” in configura5on space (where the wavefunc5on is posi5ve, say) that cannot be reached from other regions of configura5on space (with posi5ve wavefuncton) without crossing nodal surfaces. But for a more generic fermion wavefunc5on (with interac5ons, say), any configura5on with a posi5ve wavefunc5on, say, can be accessed from any other configura5on with a posi5ve wavefunc5on, without passing thru a nodal surface. Bert Halperin suggested that this laAer situa5on was perhaps not surprising, since the configuta5on space in ques5on is of such high dimension (eg 2N dimensional for N fermions in 2d).

Part 2: Majorana Fermions in superconduc5ng nanowires: Interac5on and fluctua5on effects

31

MPA Fisher, with Jason Alicea, Roman Lutchyn, Lukasz Fidkowski, Miles Stoudenmire, Oleg Starykh, Gil Refael, Felix von Oppen, Yuval Oreg, Chetan Nayak

Explore Majorana zero modes in 1d nanowires: •  Non-‐Abelian braiding sta5s5cs in 1d? •  Effects of interac5ons on 1d topo sc phase? •  Majorana Q-‐bits for sc with quasi-‐LRO? •  1d SC-‐LuJnger liquid juncLons

Simons Symposia, Quantum Physics Beyond simple systems US Virgin Islands, Jan 31, 2012

Desperately Seeking Majorana Fermions

32

Why Majorana’s?

Because they are cool, useful and might actually exist!

Majorana Vor5ces in 2d p+ip superconductors

B-‐deGennes Hamiltonian:

Vortex binds Majorana zero mode Branch cut

33

Two vor5ces makes one q-‐bit

Define complex fermion:

N vor5ces, N/2 Q-‐bits, degeneracy

1

Vortex Two vor5ces have 2 an5commu5ng Majorana operators

Non-‐Abelian Braiding Sta5s5cs

34

Each Majorana “sees” other vor5ces as carrying pi flux, introduce branch cuts

Branch cuts

Under exchange, one Majorana crosses the branch cut of the other

Unitary operator implements exchange:

Non-‐Abelian opera5on

Majorana’s in one-‐dimension?

35

Kitaev model: 1d 5ght binding model for spinless Fermions

Introduce Majorana operators

Soluble point:

Remarkably, absent from H are:

Majorana zero energy end modes

36

Non-‐local zero energy Q-‐bit:

Topological sc

Non-‐topological sc

Non-‐topological sc:

Fully gapped, no Majorana end modes

Moving/crea5ng 1d Majorana’s

37

Vary parameters: create topo to non-‐topo sc boundary – Majorana’s “live” at bondaries

Majorana can be created, moved and then annihilated (using “piano key gates”)

But can they be braided?

Lutchyn, Sau, Das Sarma (2010) Oreg, Refael, von Oppen (2010)

Braiding 1d Majorana Fermions

38

Alicea, Oreg, Refael, von Oppen, MPAF (2011) Use 1d wire networks

T-‐junc5on:

But what is their braiding sta5s5cs?

Same as vor0ces in 2d SC’s!!

Choose H real. Exchange 2 Majorana’s avoiding Pi-‐junc5ons (arrows, deno5ng sign of Delta, must be one-‐in/one-‐out). Since real, no Berry’s phase. Must bring H back to original form, mul5plying Fermion crea5on operators by i;

Bert Halperin men5oned his recent work exploring three-‐dimensional networks of p-‐wave superconduc5ng wires. Just as for the T-‐junc5on and other planar networks of wires where the exchange sta5s5cs of Majorana’s are non-‐Abelian, Majorana’s confined to 3d wire networks sa5sfy non-‐Abelian braiding sta5s5cs, modified somewhat due to the 3d geometry.

1d Experimental Systems?

40

Spin-‐orbit coupled semiconduc5ng nanowires on s-‐wave superconductor Lutchyn, Sau, das Sarma (2010) Oreg, Refael, von Oppen (2010) Edges of 2d topological insulators with sc and FM deposited on top L. Fu and C. Kane (2009) 3D topological insulator nanoribbons, A. Cook, M. Franz (2011) Cold atomic gases, L. Jiang, J. Alicea et. al. PRL (2011)

Semiconductor Nanowire

41

Nanowire, Rashba s.o. coupling, B field, SC proximity effect

Delta = 0; B-‐field opens gap in spin-‐orbit shined bands making a “helical wire”, with nonzero Delta maps to Kitaev model

Lutchyn, Sau, Das Sarma, Oreg, Refael, von Oppen

Phase diagram:

0

Vz

Topo

Non-‐topo

Topological SC:

Non-‐topo SC

Many Challenges for 1d nanowires

42

Disorder effects Interac5on effects -‐ 1d so have Lu_nger liquid Ga5ng wire on top of sc?

(1)  Interac5on Effects

(2)  Nanowire in contact with 1d superconductor (easier to gate) 3) SC-‐Lu_nger liquid junc5ons

To look at:

Interac5on Effects

43

Inside topo sc, fully gapped, no qualita5ve effects of interac5ons But, quan5ta5ve and qualita5ve changes in stability of topo sc phase (ie. phase boundary, gaps etc)

Analyze with DMRG, Hartree-‐Fock and Bosoniza5on

La_ce H with Hubbard U:

M. Stoudenmire, J. Alicea, O. Starykh and MPAF (2011)

Detec5ng 1d topo SC with DMRG

44

(1) Energy difference between ground state in even/odd Fermion parity sectors (open b.c.)

(2) Degeneracy in the “energies” of entanglement “Hamiltonian” Li and Haldane (2008)

DMRG phase diagram

45

Hubbard U shins \mu Minimum Vz to stabilize topo sc phase decreases with increasing U (good, since disturbs the bulk sc less) At larger Vz topo phase occurs over larger window of \mu (good, more robust against disorder fluctua5ons)

Physics: Interac5ons enhance effec5ve Zeeman spli_ng, and suppress the proximity sc gap

Quantum Fluctua5on Effects

46

Ques0on: Can Majorana zero modes persist if bulk SC replaced by 1d SC with quasi-‐LRO?

Fidkowski, Lutchyn, Nayak, MPAF (2011) Sau, Halperin, Flensberg, Das Sarma (2011)

Mo5va5on: Hard to gate nanowire on bulk SC, Interes5ng theore5cally First: Recover Majorana zero modes via Bosoniza0on when wire is proximate to bulk SC

Two states

Leo Kouvenhoven asked: In order to study the stability of Majorana fermions in semiconduc5ng wires in proximity to 1d (rather than 3d) superconductors, if it sufficient to consider the effects of phase fluctua5ons alone? MPAF answers: No, it is essen5al to incorporate quantum phase slip processes which lead to an energy spli_ng of two Majorana zero modes which varies as an inverse power law of their spa5al separa5on. Thus, the Majorana zero modes are not exponen5ally protected in this 1d case.

Iden5fy Majorana operators w/ Bosoniza5on

48

Two states

Fermion parity

Phase operator

Pseudo spin 1/2 operators:

An5-‐commute:

Zero energy state

Majorana operators:

Zero energy

Ady Stern asked: Are the two states, phi=0 and phi=pi, eigenstates of fermion party in the wire? MPAF answer: No. But the linear superposi5ons, phi=0 +/-‐ phi=\pi are fermion parity eigenstates.

Tunnel spli_ng of Majorana’s via Bosoniza5on

50

Bulk s-‐wave sc

vortex

Tunnel vortex “between” nanowire and bulk SC

Vortex tunneling shins phase of electron by pi

Q-‐bit spli_ng: vortex tunneling rate

Vortex tunneling is equivalent to Majorana tunneling between ends

Nanowire Proximate to 1d SC with Quasi-‐LRO

51

51

Nanowire (spinless Lu_nger liquid)

AArac5ve interac5ons in 1d SC: spin gap

1d SC: spinful electrons

Interwire Cooper pair tunneling

Degeneracy?

52

Two states (separated by a large barrier)??

No; Can let, with no cost in ac5on

Global U(1) charge symmetry (in both wires) is unbroken (no ODLRO). Electron number is certain so total phase fluctuates. Can compensate phase shin in nanowire by shining phase in 1d SC

No ground state degeneracy

Two wires proximate to 1d sc?

53

L

Is the Q-‐bit exponen5ally protected??

Cooper pair tunneling

Pseudo spin 1/2 Phase operator

Fermion parity (wire 1, say)

Q-‐bit, zero energy

Q-‐bit spli_ng: Electron tunneling

54

L

Electron tunnels between nanowires, “under” the spin gap in 1d SC

Splits the 2-‐fold degeneracy:

Exponen0al protec0on!

Q-‐bit spli_ng: Vortex tunneling

55

L

BackscaAer electron in 1d SC: quantum tunneling of vortex

Only power law protected!

Vortex shins electron phase in wire 2 by \pi:

Energy spli_ng

Various spli_ngs for 2-‐wire Q-‐bit

56

L

A B

C

A: Electron tunneling between 2 nanowires

B: Phase slip between nanowire and 1d SC

C: Phase slip in 1d SC

Leo Kouvenhoven asked: Is an electron backscaAering process in a 1d superconduc5ng wire the same as a quantum phase slip? MPAF answer: Yes. Precisely the same. Indeed, an impurity that backscaAers an electron absorbs a momentum 2kF which degrades the current. And this has the the same effect as a quantum phase slip process Leo Kouvenhoven asked: What is the meaning/value of Krho? MPAF answers: Essen5ally, Krho equals the number of channels in the (quasi-‐)1d superconduc5ng wire.

Put on the Mustard?

58

Charlie Marcus suggests “squir5ng” SC onto surface of nanowire

Virtue: More easily gate electron density in nanowire

Drawback: Lose exponen5al protec5on of Q-‐bits due to quasi-‐ODLRO

Large power possible since Krho propor5onal to number of channels in 1d sc (the mustard)

SC-‐Normal junc5on in Nanowire

59

Bulk s-‐wave SC Helical nanowire

SC-‐Normal junc5on

Non-‐interacLng electrons in Helical Wire: only 2 possibili5es (at kT=eV=0)

Perfect normal reflec5on: Perfect Andreev reflec5on:

Ordinary 1d SC Topological 1d SC Helical metal Helical metal

Presence of Majorana Fermion reflected in zero bias conductance of juncLon

Detect Majorana’s?

Charlie Marcus asked: How important are Lu_nger liquid effects? MPAF answer: Well, there are four cases to consider, a normal wire which is either helical or not, and a superconductor which is either topological or not. Weak repulsive interac5ons play a qualita5vely important role for a junc5on between a non-‐helical wire and a non-‐topological superconductor, driving the conductance to zero. Strong interac5ons can make qualita5ve differences in all four cases. Sankar Das Sarma asked: What role does the finite length of the wire play? MPAF answer: For voltages and temperatures above a length dependent crossover not much. But at lower voltages and temperatures the nature of the Fermi liquid leads will cut off the Lu_nger liquid effects. Charlie Marcus asked: Does the presence of a zero bias anomaly cons5tute an observa5on of a Majorana fermion? MPAF answer: Indirectly. If a (repulsively interac5ng) single channel wire contacts a topological superconductor with a Majorana zero mode, the wire “eats” the Majorana which necessarily leads to a zero bias conductance peak of G=2e2/h. For a non-‐topological superconductor with no Majorana, the conductance vanishes at zero bias.

SC-‐Lu_nger Liquid junc5on

61

Turn on interac5ons in Helical nanowire: Lu_nger parameter g

Ordinary 1d SC Topological 1d SC Helical LL Helical LL

g=1 non-‐interac5ng g<1 Repulsive interac5ons g>1 AArac5ve interac5ons

For 1d topo sc, perfect Andreev survives for g>1/2

Even for 1d ordinary sc, Andreev stable for g>2

Ordinary 1d SC Helical LL

“Emergence” of Majorana zero mode at juncLon

Fidkowski, Lindner, Alicea, Lutchyn, Fendley, MPAF (in prepara5on)

I-‐V Curves for 1d SC-‐LL Junc5ons

62

dI/dV

V

2e2/h

0

Ordinary 1d SC Helical LL

Topological 1d SC Helical LL

1d Topological SC-‐LL junc5on

1d Ordinary SC-‐LL junc5on

“Detects” presence/absence of Majorana zero mode

Zero bias tunneling peak

63

Conclusions: •  1d wire networks, Majorana’s sa5sfy non-‐Abelian sta5s5cs

•  On 3d SC, interac5ons in nanowire suppress gap, but reduce threshold B field and widen chem. poten5al window for aAaining topo sc phase

•  2 nanowires coupled to 1d SC (with quasi-‐ODLRO), support degenerate Q-‐bit, with power law spli_ng

•  1d SC-‐Lu_nger liquid junc5ons – “detects” presence of Majorana fermions

Theory: •  Braiding Majorana’s with quasi-‐ODLRO? •  Other non-‐Abelian par5cles possible in 1d wire networks? •  Other topological phases exist/useful? Expt: •  Observe Majorana’s in nanowires: zero bias tunneling peak, 4pi Josephson effect •  Move/braid Majorana’s •  Build topological quantum computer!

Open issues

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