発表ファイル 数理物理・物性基礎論セミナー Watanabe 1 intro

nonrelativistic systems

Haruki Watanabe

Ph.D. candidate (Vishwanath Group)

University of California, Berkeley

Appointed as a Pappalardo fellow at MIT (2015–2018)

Yoichiro Nambu (born Jan. 1921) Jeffrey Goldstone (born Sept. 1933)

1. Introduction

Main issues to be clarified today

When several internal symmetries are spontaneously broken, - what’s the number and dispersion of Goldstone modes in general? - Is this fixed by symmetry alone?

- Simple and self-contained description?

Why are they important?

- Spontaneous symmetry breaking is ubiquitous phenomenon in nature.

- Universal outcomes irrespective of microscopic details (we’ll see examples in Sec. 2) - Gapless modes determines low-energy properties of the system:

Debye’s C_{HTL µ T}^d law, Bloch’s D MzHTL µ T^{d2 law,}

…

Review:

T. Brauner, “Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems”, Symmetry 2010, 2, 609.

If time permits, I’ll also talk about

- impossiblity of “quantum time crystals”

- non-Fermi liquids due to interactions with NGBs.

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What is spontaneous symmetry breaking?

Ref: Koma-Tasaki, 1993, 1994, see also the recent lecture note.

G: Symmetry of “Hamiltonian H + commutation relations” or “Lagrangian L”® The laws of physics (EOMs, …) H: Symmetry of the state realized in nature.

When H Ì G (and H ¹ G), we say the original symmetry G is spontaneously broken into H.

G”H (element of G but not of H): broken symmetries.

The coset space GH represents the set of degenerate ground states.

UH1L8e< = S^{1, SO}H3LSOH2L = S² If G is a Lie group, G = e^{i ΕiQi} discrete subgroups. - dim G = the number of generators of G,

- dimHGHL = dim G - dim H = the number of broken symmetries dimHML: the dimension of manifold M.

dimHGHL is fixed by the continuous part of G and H (discrete subgroups do not change it).

Ÿ Definition of SSB

Order parameter ΦHxL: transforms under U Î G: Φ 'HxL = U¾ Φ HxL U.

If XΦHxL\ changes value before and after the symmetry transformation, i.e., XΦ' HxL\ = XU¾ Φ HxL U\ ¹ XΦHxL\^,

we say the symmetry is broken.

For Lie group U_{HΕL = e}^{ä Ε Q}, we use the infinitesimal form, X∆ΦHxL\ ºXΦ' HxL\-XΦHxL\

Ε ⁼X@ä Q, ΦHxLD\ ¹ 0. ¬ definition in field theory textbook. Q itself contains IR divergence but the volume integral in

X@ä Q, ΦHy, tLD\ = Ù d^{dx äYAj0}Hx, tL, ΦHy, tLE]

is harmless thanks to the locality of the commutation relation.

Ÿ ^Subtlety

Can we understand the infinite system with SSB as the thermodynamic limit of a finite-size system?

In a finite-size system,

@H, UD = 0 Þ simultaneous diagonalization H _{n, u\ = E}n _{n, u\,} U _{n, u\ = ã}^{ä u} _{n, u\.} Then,

Xn, u¤ Φ ' HxL  n, u\

=Xn, u¤ U¾ Φ HxL U  n, u\

=_{Xn, u¤ ã}^{-ä u}Φ_{HxL ã}^{ä u} _{n, u\}

=Xn, u¤ Φ HxL  n, u\

No SSB in the ground state or in the ensemble!!

D = 0 Þ   q\   q\

  \

Þ H

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No SSB in the ground state or in the ensemble!!

For Lie group,

@H, QD = 0 Þ H  n, q\ = En n, q\, ^Q n, q\ = q  n, q\. Then,

Xn, u¤@Q, ΦHxLD  n, q\

=Xn, q¤ Q ΦHxL  n, q\ - Xn, q¤ ΦHxL Q  n, q\

= qXn, q¤ ΦHxL  n, q\ - q Xn, q¤ ΦHxL  n, q\

= 0.

- This argument cannot be applied to nonabelian symmetries Þ ferromagnet is an exception.

- In general, if the order parameter does not commute witht H, no SSB in the ground state or ensemble in a finite-size system. Naive thermodynamic limit does not work.

e.g., Marshall-Lieb-Mattis theorem for Heisenberg antiferromagnet (The GS is an S = 0 state).

Ÿ Example: transverse field Ising model H = -_ÚiΣ_i^zΣ_i+1^z- g_ÚiΣ_i^x.

 gs\ = ^­\+ ¯\

2 ^{+ OHgL, UZ2} gs\ = + gs\  1 st\ = ^­\- ¯\

2

+ O_{HgL, UZ2} 1 st\ = - 1 st\ EGS- E1 stµ g^-N.

 gs\,  1 st\ are unphysical in the sense that they - do not have the clustering property

Xgs¤ Σi^zΣj^z gs\ = 1 + OHhL for i ¹ j but Xgs¤ Σi^z gs\ = 0. - are not ergodic (O(1) fluctuation of m = ^Úi=1NΣiz

N ⁾

- are not robust against a local mesurement.

Ÿ Resolution: symmetry breaking field H ' = H - h_{Ù d}^dx j_HxL

limh®+0limV®¥XΦ' HxL\ ¹ lim^{h®+0limV®¥}XΦHxL\ limh®+0limV®¥X@Q, ΦHxLD\

Note: limh®0limV®¥^{¹ lim}V®¥limh®0

The external field picks up a symmetry breaking state. e.g. H = -_ÚiΣ_i^zΣ_i+1^z- g_ÚiΣ_i^x- h_ÚiΣ_i^z

 gs\ H0L > ^­\+ ¯\

2

 gs\ HhL >  ­\

D EHhL > h N + OIg^-NM ® Even if OH1L energy cost, symmetry breaking state wins.

Ÿ Long-range order

XΦHxL ΦHyL\ ® Σ²> 0 for a large enough  x - y¤ (much larger than any microscopic length scale). limV®¥

YF²] V^{2 = Σ}

2 _{for F =}

Ù d^{dx ΦHxL.}

Note: we do not apply symmetry breaking field in the definition of Σ². e.g.  _{gs\ >} ^­\+ ¯\

2 ^.

-  ­\ and  ¯\ have a completely different expectation value of order parameter Þ cancellation after sum. -  ­\ and  ¯\ have the same long-range order Þ no cancellation.

Ÿ Kaplan, Horsch & von der Linden (1989)

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LRO Þ SSB

at T = 0 (Kaplan, Horsch & von der Linden) at T > 0 (Griffiths, Koma-Tasaki)

XgsHhL¤ F  gsHhL\

V ^³

X0¤ F^`F` 0\ V^{2 + OJ}

1 V^2N.

Variational principle:

XgsHhL¤ HH - h FL  gsHhL\ £ XΨ¤ HH - h FL  Ψ\ for any state  _Ψ\.

XgsHhL¤ F^` gsHhL\

V ^³

XΨ¤ F^` Ψ\

V ⁺

XgsHhL¤ H  gsHhL\-XΨ¤ H  Ψ\

h V ^³

XΨ¤ F^` Ψ\

V ⁺

E₀-_{XΨ¤ H  Ψ\} h V

Take  _{Ψ\ =}

 gsH0L\+^F

` gs\

³ F^` gs\·

2 ^{. then} XΨ¤ F^` Ψ\

V ⁼

Xgs¤ F^`F` gs\ V^{2 and}

XΨ¤ H  Ψ\ = Xgs¤ H  gs\ +^{Xgs¤ F}

` H F<sup>`</sup> _gs\

2_{Xgs¤ F}^{`</sup>F<sup>`} _gs\ ^{= E0+ OI} 1 V^M

Ÿ Horsch & von der Linden (1988)

LRO at T = 0 Þ V²Σ²=Xgs¤ F F  gs\ for F = Ù d^{dx ΦHxL.} XH\ =Xgs¤ F H F  0\

X0¤ F F  0\ ^{= E0+} 1 2

Xgs¤@F,@ H, FD  gs\

Xgs¤ F F  gs\ ^{= E0+ OI} 1

VM where H  gs\ = E⁰ gs\. - F 0\ is a low-lying energy state.

- Ortogonal to the ground state: Xgs¤ F  gs\ = 0.

Ÿ Divergence of susceptibility Χ ³ ^IV

2_Σ2_M²

X0¤BF^{`,</sup>BH,F<sup>`}FF  0\^{µ V} 2_.

* V²Σ²=_{Xgs¤ F}<sup>`</sup>F<sub>`  gs\ = Ún¹0Xgs¤ F</sub>`  n\Xn¤F`  gs\ =Ù d Ω SHΩL^, where S_{HΩL = Ún¡Xgs¤ F`  n\¥}²∆_{HΩ - EnL.}

* Perturbation theory:  _gs\^H1L= _{gs\ + h}  n\ Xn¤ F  gs\

En ^{HE0= 0L}

HXgs¤ F  gs\L^H1L

V ⁼

1

V ^{JXgs¤ + h} Xgs¤ F  n\

Ω_n Xn¤N FJ gs\ + Ε  n\^{Xn¤ F  gs\}_Ω

n ^{N =}

Xgs¤ F  gs\

V ^{+ h}

2 V

Xgs¤ F  n\ Xn¤ F  gs\

Ω_n ^{+ OIh}

2_M

Χ =

HXgs¤ F  gs\L^H1L

V ^-

Xgs¤ F  gs\ V

h ⁼

2 V

Xgs¤ F  n\ Xn¤ F  gs\

E_n ⁼

2

V Ù d Ω^SHΩLΩ

* ¹

2^X0¤AF

`,<sub>AH, F</sub>` EE 0\ = Ún¡X0¤ F`  n\¥^2En=Ù d Ω Ω SHΩL Cauchy–Schwarz inequality _Hx×yL²£Hx×xL Hy×yL I_{Ù d Ω SHΩL}M^2£IÙ d Ω HΩ SHΩLLMIÙ d Ω' HSHΩ'LΩ'L^M

IV^2Σ2M^2£ ₂¹X0¤AF<sup>`,</sup>AH, F` EE 0^ _V^{2 Χ} I took this argument from

L. Capriotti, Int. J. Mod. Phys. B (2001)

(Anderson) Tower of states

Review: C. Lhuillier, cond-mat/0502464

Some exact results in Koma-Tasaki, J. Stat. Phys. 76, 745 (1994)

Ÿ ^Crystals H = ¹

2^mÚix

i²-_{Úi< j}V_Ixi- xjM

= ^P2

2 m N^+Úk¹0B p_k¾×p_k

2 m n₀ ^{+ EHkiukLF}

.

where P is the total momentum of the system. The ground state is a plane-wave state P = 0.

The observed crsytal (symmetry-breaking state) should be a superposition:

 crystal\ = Ú^GCG P = G\ ® higher energy than the ground state at least in a finite volume system. D E = ^YP

2_]

2 m N ^=ÚG CG¤ 2 ^G2

2 m N ^µ 1 Volume^?

If _YP²] = OHVL in the “physical” symmetry-breaking state, D E remains OH1L even after taking V ® ¥ limit (?).

Ÿ Antiferromangets L₌ ^Ρ

2 v^{2 n}

  × n  - ¹

2 ^{Ρ Ñin × Ñin,}

H = ^v2

2 Ρ S×S

V ^+Úk¹0J v²

2 Ρ^{sk¾× sk+} 1 2 ^{Ρ k}

2_n k^{¾× n}kN where S =_{Ù d x} ^Ρ

v²ⁿ

 ´ n. AsΑ, sΒE = i ΕΑΒΓsΓ

As^{Α, nΒ}E = i Ε^ΑΒΓnΓ An^{Α, nΒ}E = 0.

Neel order n and the uniform magnetization s do not commute ® if we fix s, n is uncertain.

From Lhuillier’s review

Ÿ ^BEC

H = i Ψ¾ Ψ  - ¹

2 mÑ Ψ¾ × Ñ Ψ + Μ Ψ¾ Ψ - ^g

2^{HΨ¾ ΨL} 2

=Μ^HN-N0L2

2 N₀ ^+Úk¹0J Μ k²

2 m g^Θk¾Θk+ g 2^{J1 +}

k²

4 m Μ^{N ∆ nk¾∆ nkN}

.

Coherent state ® _{Y∆ N}²_{] = N}0 and XH\ = ̐2 = OH1L. ref: Shimizu-Miyadera PRD (2001)

Ÿ Ferromangets

L_{= m}HcosΘ - 1L Φ  ⁺_{2 v}^Ρ2ⁿ

 × n , H =_Úk¹

2 ^{Ρ k} 2_n

k^{¾× n}k⁼_Úk¹0 1 2 ^{Ρ k}

2_n k^{¾× n}k^.

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Ú ₂ ^¾ Ú ₂ ^¾ No TOS! No sutlety.

Nambu-Goldstone theorem for Lorentz-invariant systems

九 後汰一郎 (Taichro Kugo)「 ゲージ場 の 量子 論II」

Ÿ ^Statement

Ÿ ^Proof

To derive the simple spectral rep, one has to use the Lorentz invaiance - _{Xn, k¤ j}^Р_{n, k\ µ k}^Μ

- UL n, k\ µ  n, L k\ ¬ P^Μ n, k\ = k^Μ n, k\ - DF_{Ix, Σ}²_{M = Ù}

d⁴k ä_{H2 ΠL}⁴

1 Σ²-k²-ä Ε^ã

-ä k x

Ÿ Number of NGBs

Suppose jΜ¹= fΠ¹¶ΜΦ¹, jΜ²= fΠ²¶ΜΦ². Lorentz invariance ® fΠ^1,2 are real. If Φ¹= Φ²= Φ, then

fΠ²jΜ¹- fΠ¹jΜ²= fΠ²_IfΠ¹¶ΜΦ_{M - f}Π¹_IfΠ²¶ΜΦ_{M = 0} Þ fΠ²Q¹- fΠ¹Q² is unbroken (contradiction).

Nambu-Goldstone theorem for Lorentz-invariant systems

Review: T. Brauner, “Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems”, Symmetry 2010, 2, 609

X0¤@QVHtL, ΦH0LD  0\

=_ÙVd^dxX0¤@jHx, tL, ΦH0LD  0\

=_Ù

V^d d_xAÚ

nX0¤ jH0L  n\ Xn¤ ΦH0L  0\ ã^{ä kn×x-ä Ent-}ÚnX0¤ ΦH0L  n\ Xn¤ jH0L  0\ ã^{-ä kn×x+ä Ent}E

=_{ÚnH2 ΠL}^d∆_kn,0AX0¤ jH0L  n\ Xn¤ ΦH0L  0\ ã^{-ä Ent-}X0¤ ΦH0L  n\ Xn¤ jH0L  0\ ã^{+ä Ent}E Note that the ground state  n\ =  0\ does not contribute to the sum:

X0¤ jH0L  0\ X0¤ ΦH0L  0\ - X0¤ ΦH0L  0\ X0¤ jH0L  0\ = 0.

We also note that _X0¤@QVHtL, ΦH0LD  0\ is (i) time independent and (ii) nonzero.

Hence, some excitation  NG\ with ENGHk = 0L = 0 should couple to the current X0¤ jH0L  NG\ ¹ 0 and the order parame- ter XNG¤ ΦH0L  0\ ¹ 0.

- Not necessarily modes? - How many NGBs? - What’s the disperison?

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